A sequence $S _ { 1 } , S _ { 2 } , S _ { 3 }$ ... is deﬁned recursively as follows: \[\begin{array} { l } s _ { k } = 5 s _ { k - 1 } + \left( s _ { k - 2 } \right) ^ { 2 } \quad \text { for all integers } k \geq 3 \\ s _ { 1 } = 4 \\ s _ { 2 } = 8 \end{array}\] Use (strong) mathematical induction to prove that sn is divisible by 4 for all integers \[n \geq 1\]

Proof (by strong mathematical induction)

A sequence $a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$ is defined recursively as follows: $\begin{array} { l } a _ { 0 } = 2 , \quad a _ { 1 } = 9 \\ a _ { k } = 5 a _ { k - 1 } - 6 a _ { k - 2 } \text { for all integers } k \geq 2 \end{array}$ Use strong mathematical induction to prove that for all integers \[n \geq 0 , a _ { n } = 5 \cdot 3 ^ { n } - 3 \cdot 2 ^ { n }\]

Proof (by strong mathematical induction)

Define a set $S$ recursively as follows: I. BASIS: $\epsilon \in S$ II. RECURSION: If $s$ and $t$ are in $S$, then $\begin{array} { l l l l } \text { a. } 0 s \in S & \text { b. } s 0 \in S & \text { c. } 1 s 1 t \in S & \text { d. } s 1 t 1 \in S \end{array}$ III. RESTRICTION: No strings other than those derived from I and II are in $S$. Use structural induction to prove that every string in $S$ contains an even number of l's.

@#LAT-DLM& tains an even number of 1's. Show that

Exam 5: Sequences, Mathematical Induction, and Recursion

A sequence $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$ satisfies the recurrence relation $b _ { k } = 2 b _ { k - 1 } + 8 b _ { k - 2 }$ with initial conditions $b _ { 1 } = 1$ and $b _ { 2 } = 0$ . Find an explicit formula for the sequence.

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Question 1

Correct Answer:

Verified

The characteristic equation is $t ^ { 2 } - 2 t - 8 = 0$ . Since $t ^ { 2 } - 2 t - 8 = ( t - 4 ) ( t + 2 )$ , there are two roots: $t = 4$ and $t = - 2$ By the distinct roots theorem, there exist constants $C$ and $D$ such that
$b _ { n } = C \cdot 4 ^ { n } + D \cdot ( - 2 ) ^ { n } \quad$ for all integers $n \geq 0 .$
Since $b _ { 0 } = 1$ and $b _ { 1 } = 0$ , then
$\begin{array} { l } \left\{ \begin{array} { l } b _ { 0 } = C \cdot 4 ^ { 0 } + D \cdot ( - 2 ) ^ { 0 } = C + D = 1 \\b _ { 1 } = C \cdot 4 ^ { 1 } + D \cdot ( - 2 ) ^ { 1 } = 4 C - 2 D = 0\end{array} \right\} \Leftrightarrow \left\{ \begin{array} { l } C = 1 - D \\D = 2 C\end{array} \right\} \\\Leftrightarrow \left\{ \begin{array} { l } C = 1 - 2 C \\D = 2 C\end{array} \right\} \Leftrightarrow \left\{ \begin{array} { l } C = 1 / 3 \\D = 2 / 3\end{array} \right\}\end{array}$
Thus $b _ { n } = \frac { 1 } { 3 } \cdot 4 ^ { n } + \frac { 2 } { 3 } \cdot ( - 2 ) ^ { n }$ for all integers $n \geq 0$ .

Use mathematical induction to prove that for all integers $n \geq 1$ , $4 + 8 + 12 + \cdots + 4 n = 2 n ^ { 2 } + 2 n .$

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Question 2

Correct Answer:

Verified

$\underline { \text { Proof (by mathematical induction) } }$ : Let the property $P ( n )$ be the equation
$4 + 8 + 12 + \cdots + 4 n = 2 n ^ { 2 } + 2 n . \quad \leftarrow P ( n )$
Note that $4 + 8 + 12 + \cdots + 4 n = \sum _ { i = 1 } ^ { n } 4 i$ Show that $P ( 1 )$ is true: $P ( 1 )$ is true because the left-hand side is $\sum _ { i = 1 } ^ { 1 } 4 i = 4 \cdot 1 = 4$ and the right-hand side is $2 \cdot 1 ^ { 2 } + 2 \cdot 1 = 4$ also.

Show that for all integers $k \geq 1$ , if $P ( k )$ is true then $P ( k + 1 )$ is true: Let $k$ be any integer with $k \geq 1$ , and suppose that
$4 + 8 + 12 + \cdots + 4 k = 2 k ^ { 2 } + 2 k . \quad \leftarrow \quad \begin{array} { l } P ( k ) \\\text { inductive hypothesis }\end{array}$
We must show that
$4 + 8 + 12 + \cdots + 4 ( k + 1 ) = 2 ( k + 1 ) ^ { 2 } + 2 ( k + 1 ) . \quad \leftarrow P ( k + 1 )$
Now the left-hand side of $P ( k + 1 )$ is
$\begin{aligned} 4 + 8 + 12 + \cdots + 4 ( k + 1 ) = & 4 + 8 + 12 + \cdots + 4 k + 4 ( k + 1 ) \\ = & \text { by making the next-to-last term explicit } \\ & \left( 2 k ^ { 2 } + 2 k \right) + 4 ( k + 1 ) \\ & \text { by inductive hypothesis } \\ & = \begin{array} { c } 2 k ^ { 2 } + 6 k + 4 \\ \text { by multiplying out and combining like terms. } \end{array} \end{aligned}$
And the right-hand side of $P ( k + 1 )$ is
$2 ( k + 1 ) ^ { 2 } + 2 ( k + 1 ) = 2 \left( k ^ { 2 } + 2 k + 1 \right) + 2 k + 2 = 2 k ^ { 2 } + 4 k + 2 + 2 k + 2 = 2 k ^ { 2 } + 6 k + 4 .$
Thus the left-hand and right-hand sides of $P ( k + 1 )$ are equal (as was to be shown).

For each integer $n \geq 0$ , let $P ( n )$ be the equation $For each integer n \geq 0 , let P ( n ) be the equation (Recall that by definition 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \sum _ { i = 0 } ^ { n } 3 ^ { i } ) (a) Is P(0) true? Justify your answer. (b) In the inductive step of a proof that P ( n ) is true for all integers n \geq 0 , we suppose P ( k ) is true (this is the inductive hypothesis), and then we show that P ( k + 1 ) is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation. Proof that for all integers k \geq 0 , if P ( k ) is true then P ( k + 1 ) is true: Let k be any integer that is greater than or equal to 3 , and suppose that We must showthat (c) Finish the proof started in (b) above.$ (Recall that by definition $1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \sum _ { i = 0 } ^ { n } 3 ^ { i }$ ) (a) Is P(0) true? Justify your answer. (b) In the inductive step of a proof that $P ( n )$ is true for all integers $n \geq 0$ , we suppose $P ( k )$ is true (this is the inductive hypothesis), and then we show that $P ( k + 1 )$ is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation. Proof that for all integers $k \geq 0$ , if $P ( k )$ is true then $P ( k + 1 )$ is true: Let $k$ be any integer that is greater than or equal to 3 , and suppose that $For each integer n \geq 0 , let P ( n ) be the equation (Recall that by definition 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \sum _ { i = 0 } ^ { n } 3 ^ { i } ) (a) Is P(0) true? Justify your answer. (b) In the inductive step of a proof that P ( n ) is true for all integers n \geq 0 , we suppose P ( k ) is true (this is the inductive hypothesis), and then we show that P ( k + 1 ) is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation. Proof that for all integers k \geq 0 , if P ( k ) is true then P ( k + 1 ) is true: Let k be any integer that is greater than or equal to 3 , and suppose that We must showthat (c) Finish the proof started in (b) above.$ We must showthat $For each integer n \geq 0 , let P ( n ) be the equation (Recall that by definition 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \sum _ { i = 0 } ^ { n } 3 ^ { i } ) (a) Is P(0) true? Justify your answer. (b) In the inductive step of a proof that P ( n ) is true for all integers n \geq 0 , we suppose P ( k ) is true (this is the inductive hypothesis), and then we show that P ( k + 1 ) is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation. Proof that for all integers k \geq 0 , if P ( k ) is true then P ( k + 1 ) is true: Let k be any integer that is greater than or equal to 3 , and suppose that We must showthat (c) Finish the proof started in (b) above.$ (c) Finish the proof started in (b) above.

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Question 3

Correct Answer:

Verified

For each integer $n \geq 0$ , let $P ( n )$ be the equation
$For each integer n \geq 0 , let P ( n ) be the equation (Recall that by definition 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \sum _ { i = 0 } ^ { n } 3 ^ { i } ) (a) P ( 0 ) : \sum _ { i = 0 } ^ { 0 } 3 ^ { i } = \frac { 3 ^ { 0 + 1 } - 1 } { 2 } P ( 0 ) is true because the left-hand side equals \sum _ { i = 0 } ^ { 0 } 3 ^ { i } = 3 ^ { 0 } = 1 , and the right-hand side equals \frac { 3 ^ { 0 + 1 } - 1 } { 2 } = \frac { 3 - 1 } { 2 } = 1 also. (b) P ( k ) : 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k } = \frac { 3 ^ { k + 1 } - 1 } { 2 } P ( k + 1 ) : 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k + 1 } = \frac { 3 ^ { ( k + 1 ) + 1 } - 1 } { 2 } , Or, equivalently, P ( k + 1 ) is 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k + 1 } = \frac { 3 ^ { k + 2 } - 1 } { 2 } . (c) Proof that for all integers k \geq 0 , if P ( k ) is true then P ( k + 1 ) is true: Let k be any integer that is greater than or equal to 0 , and suppose that 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k } = \frac { 3 ^ { k + 1 } - 1 } { 2 } . \quad \leftarrow \quad \begin{array} { l } P ( k ) \\ \text { inductive hypothesis } \end{array} We must show that 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k + 1 } = \frac { 3 ^ { k + 2 } - 1 } { 2 } . \quad \leftarrow P ( k + 1 ) Now the left-hand side of P ( k + 1 ) is 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k + 1 } = 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k } + 3 ^ { k + 1 } by making the next-to-last term explicit = \frac { 3 ^ { k + 1 } - 1 } { 2 } + 3 ^ { k + 1 } = \frac { 3 ^ { k + 1 } - 1 } { 2 } + \frac { 2 \cdot 3 ^ { k + 1 } } { 2 } = \frac { 3 \cdot 3 ^ { k + 1 } - 1 } { 2 } = \frac { 3 ^ { k + 2 } - 1 } { 2 \mathrm { by } \text { adding the fractions and combining like terms } } by a law of exponents, = \frac { 3 ^ { k + 2 } - 1 } { 2 } which equals the right-hand side of P ( k + 1 ) . Thus the left-hand and right-hand sides of P ( k + 1 ) are equal [as was to be shown].$
(Recall that by definition $1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \sum _ { i = 0 } ^ { n } 3 ^ { i }$ )
(a) $P ( 0 ) : \sum _ { i = 0 } ^ { 0 } 3 ^ { i } = \frac { 3 ^ { 0 + 1 } - 1 } { 2 }$
$P ( 0 )$ is true because the left-hand side equals $\sum _ { i = 0 } ^ { 0 } 3 ^ { i } = 3 ^ { 0 } = 1$ , and the right-hand side equals $\frac { 3 ^ { 0 + 1 } - 1 } { 2 } = \frac { 3 - 1 } { 2 } = 1$ also.
(b) $P ( k ) : 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k } = \frac { 3 ^ { k + 1 } - 1 } { 2 }$
$P ( k + 1 ) : 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k + 1 } = \frac { 3 ^ { ( k + 1 ) + 1 } - 1 } { 2 } ,$
Or, equivalently, $P ( k + 1 )$ is $1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k + 1 } = \frac { 3 ^ { k + 2 } - 1 } { 2 }$ .
(c) Proof that for all integers $k \geq 0$ , if $P ( k )$ is true then $P ( k + 1 )$ is true:
Let $k$ be any integer that is greater than or equal to 0 , and suppose that
$1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k } = \frac { 3 ^ { k + 1 } - 1 } { 2 } . \quad \leftarrow \quad \begin{array} { l } P ( k ) \\\text { inductive hypothesis }\end{array}$
We must show that
$1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k + 1 } = \frac { 3 ^ { k + 2 } - 1 } { 2 } . \quad \leftarrow P ( k + 1 )$
Now the left-hand side of $P ( k + 1 )$ is
$1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k + 1 } = 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { k } + 3 ^ { k + 1 }$
by making the next-to-last term explicit
$= \frac { 3 ^ { k + 1 } - 1 } { 2 } + 3 ^ { k + 1 }$
$= \frac { 3 ^ { k + 1 } - 1 } { 2 } + \frac { 2 \cdot 3 ^ { k + 1 } } { 2 }$ $= \frac { 3 \cdot 3 ^ { k + 1 } - 1 } { 2 }$ $= \frac { 3 ^ { k + 2 } - 1 } { 2 \mathrm { by } \text { adding the fractions and combining like terms } }$ by a law of exponents, $= \frac { 3 ^ { k + 2 } - 1 } { 2 }$
which equals the right-hand side of $P ( k + 1 )$ .
Thus the left-hand and right-hand sides of $P ( k + 1 )$ are equal [as was to be shown].

Use mathematical induction to prove that for all integers $n \geq 0$ , $1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } .$

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Question 4

A sequence $S _ { 1 } , S _ { 2 } , S _ { 3 }$ ... is deﬁned recursively as follows: =5+ for all integers k\geq3 =4 =8 Use (strong) mathematical induction to prove that sn is divisible by 4 for all integers $n \geq 1$

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Question 5

Use mathematical induction to prove that for all integers $n \geq 0,8 ^ { n } - 1 \text { is divisible by } 7 \text {. }$

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Question 6

A sequence is deﬁned recursively as follows: $s _ { k } = 5 s _ { k - 1 } + 1$ for all integers $k \geq 1$ $s _ { 0 } = 1 \text {. }$ Use mathematical induction to verify that this sequence satisﬁes the explicit formula $s _ { n } = \frac { 5 ^ { n + 1 } - 1 } { 4 } \quad \text { for all integers } n \geq 0 \text {. }$

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Question 7

For each integer $n \geq 3$ , let $P ( n )$ be the equation $3+4+5+\cdots+n=\frac{(n-2)(n+3)}{2} \cdot \leftarrow P(n)$ (Recall that by definition $3 + 4 + 5 + \cdots + n = \sum _ { i = 3 } ^ { n } i _ {. }$ ) (a) Is $P ( 3 )$ true? Justify your answer. (b) In the inductive step of a proof that $P ( n )$ is true for all integers $n \geq 3$ , we suppose $P ( k )$ is true (this is the inductive hypothesis), and then we show that $P ( k + 1 )$ is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation. Proof that for all integers $k \geq 3$ , if $P ( k )$ is true then $P ( k + 1 )$ is true: Let $k$ be any integer that is greater than or equal to 3 , and suppose that_ We must show that______ (c) Finish the proof started in (b) above.

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Question 8

A sequence $d _ { 1 } , d _ { 2 } , d _ { 3 } , \ldots$ satisfies the recurrence relation $d _ { k } = 8 d _ { k - 1 } - 16 d _ { k - 2 }$ with initial conditions $d _ { 1 } = 0$ and $d _ { 2 } = 1$ . Find an explicit formula for the sequence.

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Question 9

A sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$ satisfies the recurrence relation $c _ { k } = 6 c _ { k - 1 } - 9 c _ { k - 2 }$ with initial conditions $c _ { 0 } = 1$ and $c _ { 1 } = 6$ . Find an explicit formula for the sequence.

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Question 10

Use mathematical induction to prove that for all integers $n \geq 3$ , $2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n = \frac { ( n - 2 ) \left( n ^ { 2 } + 2 n + 3 \right) } { 3 } .$

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Question 11

A sequence $a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$ is defined recursively as follows: =2,=9 =5-6 for all integers k\geq2 Use strong mathematical induction to prove that for all integers $n \geq 0 , a _ { n } = 5 \cdot 3 ^ { n } - 3 \cdot 2 ^ { n }$

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Question 12

Use mathematical induction to prove that for all integers $n \geq 5,1 + 4 n < 2 ^ { n }$

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Question 13

Use mathematical induction to prove that for all integers $n \geq 3$ , $3 + 4 + 5 + \cdots + n = \frac { ( n - 2 ) ( n + 3 ) } { 2 } .$

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Question 14

Use summation notation to rewrite the following: $1 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 }$

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Question 15

For each integer $n \geq 3$ , let $P ( n )$ be the equation $For each integer n \geq 3 , let P ( n ) be the equation (Recall that by definition \left. 2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n = \sum _ { i = 3 } ^ { n } ( i - 1 ) \cdot i . \right) (a) Is P ( 3 ) true? Justify your answer. (b) In the inductive step of a proof that P ( n ) is true for all integers n \geq 3 , we suppose P ( k ) is true (this is the inductive hypothesis), and then we show that P ( k + 1 ) is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation. Proof that for all integers k \geq 3 , if P ( k ) is true then P ( k + 1 ) is true: Let k be any integer that is greater than or equal to 3 , and suppose that We must show that_ (c) Finish the proof started in (b) above.$ (Recall that by definition $\left. 2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n = \sum _ { i = 3 } ^ { n } ( i - 1 ) \cdot i . \right)$ (a) Is $P ( 3 )$ true? Justify your answer. (b) In the inductive step of a proof that $P ( n )$ is true for all integers $n \geq 3$ , we suppose $P ( k )$ is true (this is the inductive hypothesis), and then we show that $P ( k + 1 )$ is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation. Proof that for all integers $k \geq 3$ , if $P ( k )$ is true then $P ( k + 1 )$ is true: Let $k$ be any integer that is greater than or equal to 3 , and suppose that We must show that_ (c) Finish the proof started in (b) above.

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Question 16

A sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$ is defined as follows: $c _ { 0 } = 1 \quad \text { and } \quad c _ { k } = 7 c _ { k - 1 } + 2 \text { for each integer } k \geq 1 \text {. }$ (a) Find $c _ { 1 }$ and $c _ { 2 }$ . (b) Simplify the expression $7 ^ { n } + 2 \cdot 7 ^ { n - 1 } + \cdots + 2 \cdot 7 ^ { 2 } + 2 \cdot 7 + 2 .$ using one of the following reference formulas: $1 + 2 + 3 + \cdots + n = \frac { n ( n + 1 ) } { 2 }$ for all integers $n \geq 1$ . $1 + r + r ^ { 2 } + \cdots + r ^ { m } = \frac { r ^ { m + 1 } - 1 } { r - 1 }$ for all integers $m \geq 0$ and all real numbers $r \neq 1$ . (c) Use iteration to guess an explicit formula for the sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$

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Question 17

Transform the following summation by making the change of variable j = k + 1 : $\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } } { n }$

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Question 18

A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined as follows: $a _ { 1 } = 3 , \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 2 \quad \text { for all integers } k \geq 2 \text {. }$ (a) Find $a _ { 1 } , a _ { 2 }$ , and $a _ { 3 }$ . (b) Supposing that $a _ { 5 } = 4 ^ { 4 } \cdot 3 + 4 ^ { 3 } \cdot 2 + 4 ^ { 2 } \cdot 2 + 4 \cdot 2 + 2$ , find a similar numerical expression for $a _ { 6 }$ by substituting the right-hand side of this equation in place of $a _ { 5 }$ in the equation $a _ { 6 } = 4 \cdot a _ { 5 } + 2 .$ (c) Guess an explicit formula for $a _ { n }$ . Simplify your answer using one of the following reference formulas: $1 + 2 + 3 + \cdots + n = \frac { n ( n + 1 ) } { 2 }$ for all integers $n \geq 1$ . $1 + r + r ^ { 2 } + \cdots + r ^ { m } = \frac { r ^ { m + 1 } - 1 } { r - 1 }$ for all integers $m \geq 0$ and all real numbers $r \neq 1$ .

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Question 19

Define a set $S$ recursively as follows: I. BASIS: $\epsilon \in S$ II. RECURSION: If $s$ and $t$ are in $S$ , then a. 0s\inS b. s0\inS c. 1s1t\inS d. s1t1\inS III. RESTRICTION: No strings other than those derived from I and II are in $S$ . Use structural induction to prove that every string in $S$ contains an even number of l's.

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Question 20

Showing 1 - 20 of 37

A sequence $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$ satisfies the recurrence relation $b _ { k } = 2 b _ { k - 1 } + 8 b _ { k - 2 }$ with initial conditions $b _ { 1 } = 1$ and $b _ { 2 } = 0$ . Find an explicit formula for the sequence.

Use mathematical induction to prove that for all integers $n \geq 1$ , $4 + 8 + 12 + \cdots + 4 n = 2 n ^ { 2 } + 2 n .$

Use mathematical induction to prove that for all integers $n \geq 0$ , $1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } .$

A sequence $S _ { 1 } , S _ { 2 } , S _ { 3 }$ ... is deﬁned recursively as follows: =5+ for all integers k\geq3 =4 =8 Use (strong) mathematical induction to prove that sn is divisible by 4 for all integers $n \geq 1$

Use mathematical induction to prove that for all integers $n \geq 0,8 ^ { n } - 1 \text { is divisible by } 7 \text {. }$

A sequence $d _ { 1 } , d _ { 2 } , d _ { 3 } , \ldots$ satisfies the recurrence relation $d _ { k } = 8 d _ { k - 1 } - 16 d _ { k - 2 }$ with initial conditions $d _ { 1 } = 0$ and $d _ { 2 } = 1$ . Find an explicit formula for the sequence.

A sequence $c _ { 0 } , c _ { 1 } , c _ { 2 } , \ldots$ satisfies the recurrence relation $c _ { k } = 6 c _ { k - 1 } - 9 c _ { k - 2 }$ with initial conditions $c _ { 0 } = 1$ and $c _ { 1 } = 6$ . Find an explicit formula for the sequence.

Use mathematical induction to prove that for all integers $n \geq 3$ , $2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n = \frac { ( n - 2 ) \left( n ^ { 2 } + 2 n + 3 \right) } { 3 } .$

A sequence $a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$ is defined recursively as follows: =2,=9 =5-6 for all integers k\geq2 Use strong mathematical induction to prove that for all integers $n \geq 0 , a _ { n } = 5 \cdot 3 ^ { n } - 3 \cdot 2 ^ { n }$

Use mathematical induction to prove that for all integers $n \geq 5,1 + 4 n < 2 ^ { n }$

Use mathematical induction to prove that for all integers $n \geq 3$ , $3 + 4 + 5 + \cdots + n = \frac { ( n - 2 ) ( n + 3 ) } { 2 } .$

Use summation notation to rewrite the following: $1 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 }$

Transform the following summation by making the change of variable j = k + 1 : $\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } } { n }$

Speaking Mathematically

The Logic of Compound Statements

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Set Theory

Functions

Relations

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Exam 5: Sequences, Mathematical Induction, and Recursion

Use mathematical induction to prove that for all integers n≥1n \geq 1n≥1 , 4+8+12+⋯+4n=2n2+2n.4 + 8 + 12 + \cdots + 4 n = 2 n ^ { 2 } + 2 n .4+8+12+⋯+4n=2n2+2n.

Use mathematical induction to prove that for all integers n≥0n \geq 0n≥0 , 1+3+32+⋯+3n=3n+1−12.1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } .1+3+32+⋯+3n=23n+1−1​.

A sequence S1,S2,S3S _ { 1 } , S _ { 2 } , S _ { 3 }S1​,S2​,S3​ ... is deﬁned recursively as follows: =5+ for all integers k\geq3 =4 =8 Use (strong) mathematical induction to prove that sn is divisible by 4 for all integers n≥1n \geq 1n≥1

Use mathematical induction to prove that for all integers n≥0,8n−1 is divisible by 7. n \geq 0,8 ^ { n } - 1 \text { is divisible by } 7 \text {. }n≥0,8n−1 is divisible by 7.

Use mathematical induction to prove that for all integers n≥3n \geq 3n≥3 , 2⋅3+3⋅4+⋯+(n−1)⋅n=(n−2)(n2+2n+3)3.2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n = \frac { ( n - 2 ) \left( n ^ { 2 } + 2 n + 3 \right) } { 3 } .2⋅3+3⋅4+⋯+(n−1)⋅n=3(n−2)(n2+2n+3)​.

Use mathematical induction to prove that for all integers n≥5,1+4n<2nn \geq 5,1 + 4 n < 2 ^ { n }n≥5,1+4n<2n

Use mathematical induction to prove that for all integers n≥3n \geq 3n≥3 , 3+4+5+⋯+n=(n−2)(n+3)2.3 + 4 + 5 + \cdots + n = \frac { ( n - 2 ) ( n + 3 ) } { 2 } .3+4+5+⋯+n=2(n−2)(n+3)​.

Use summation notation to rewrite the following: 13−23+33−43+531 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 }13−23+33−43+53

Transform the following summation by making the change of variable j = k + 1 : ∑k=1nk2n\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } } { n }∑k=1n​nk2​

Speaking Mathematically

The Logic of Compound Statements

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Set Theory

Functions

Relations

Counting and Probability

Graphs and Trees

Analyzing Algorithm Efficiency

Regular Expressions and Finite State Automata

Filters

Use mathematical induction to prove that for all integers $n \geq 1$ , $4 + 8 + 12 + \cdots + 4 n = 2 n ^ { 2 } + 2 n .$

Use mathematical induction to prove that for all integers $n \geq 0$ , $1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } .$

A sequence $S _ { 1 } , S _ { 2 } , S _ { 3 }$ ... is deﬁned recursively as follows: =5+ for all integers k\geq3 =4 =8 Use (strong) mathematical induction to prove that sn is divisible by 4 for all integers $n \geq 1$

Use mathematical induction to prove that for all integers $n \geq 0,8 ^ { n } - 1 \text { is divisible by } 7 \text {. }$

Use mathematical induction to prove that for all integers $n \geq 3$ , $2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n = \frac { ( n - 2 ) \left( n ^ { 2 } + 2 n + 3 \right) } { 3 } .$

Use mathematical induction to prove that for all integers $n \geq 5,1 + 4 n < 2 ^ { n }$

Use mathematical induction to prove that for all integers $n \geq 3$ , $3 + 4 + 5 + \cdots + n = \frac { ( n - 2 ) ( n + 3 ) } { 2 } .$

Use summation notation to rewrite the following: $1 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 }$

Transform the following summation by making the change of variable j = k + 1 : $\sum _ { k = 1 } ^ { n } \frac { k ^ { 2 } } { n }$