Question 1

Find the indicated quantity.
-$a _ { 31 }$, when $\mathrm { a } _ { 1 } = 1$ and $\mathrm { d } = - \frac { 7 } { 6 }$

Accepted Answer

A)  $\frac { 223 } { 6 }$ 
B)  36 
C)  $- 34$ 
D)  $- \frac { 211 } { 6 }$ 
A)  $\frac { 223 } { 6 }$ 
B)  36 
C)  $- 34$ 
D)  $- \frac { 211 } { 6 }$

Question 2

Find the nth, or general, term.
-$25,5,1 , \ldots$

Accepted Answer

A)  $5 ^ { n - 1 }$ 
B)  $5 ^ { n }$ 
C)  $5 ^ { 3 - n }$ 
D)  $5 ( \mathrm { n } - 1 )$ 
A)  $5 ^ { n - 1 }$ 
B)  $5 ^ { n }$ 
C)  $5 ^ { 3 - n }$ 
D)  $5 ( \mathrm { n } - 1 )$

Question 3

Expand.
-$$\left( \frac { 5 } { x } - 3 x \right) ^ { 4 }$$

Accepted Answer

A)  $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 1350 x - 540 x ^ { 2 } + 81 x ^ { 4 }$ 
B)  $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 2700 - 540 x ^ { 2 } + 81 x ^ { 4 }$ 
C)  $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 3 } } + 1350 - 540 x ^ { 3 } + 81 x ^ { 4 }$ 
D)  $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 1350 - 540 x ^ { 2 } + 81 x ^ { 4 }$ 
A)  $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 1350 x - 540 x ^ { 2 } + 81 x ^ { 4 }$ 
B)  $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 2700 - 540 x ^ { 2 } + 81 x ^ { 4 }$ 
C)  $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 3 } } + 1350 - 540 x ^ { 3 } + 81 x ^ { 4 }$ 
D)  $\frac { 625 } { x ^ { 4 } } - \frac { 1500 } { x ^ { 2 } } + 1350 - 540 x ^ { 2 } + 81 x ^ { 4 }$

Question 4

Use mathematical induction to prove the following.
-$$\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \ldots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$$

Accepted Answer

Answers may vary. One possibility:
\[\be

Question 5

Use mathematical induction to prove the following.
-$$6 + 12 + 18 + \ldots + 6 n = 3 n ( n + 1 )$$

Accepted Answer

Answers may vary. One possibility: \[\be

Question 6

Find the sum, if it exists.
-$$8 - 9 + \frac { 81 } { 8 } - \frac { 729 } { 64 } + \ldots$$

Accepted Answer

A)  $- \frac { 64 } { 17 }$ 
B)  $\frac { 72 } { 17 }$ 
C)  $- \frac { 72 } { 17 }$ 
D)  Does not exist 
A)  $- \frac { 64 } { 17 }$ 
B)  $\frac { 72 } { 17 }$ 
C)  $- \frac { 72 } { 17 }$ 
D)  Does not exist

Question 7

Use mathematical induction to prove the following.
-$$\sum _ { i = 1 } ^ { n } ( 5 i + 8 ) = \frac { n ( 5 n + 21 ) } { 2 }$$

Accepted Answer

\[\begin{array} { l } 
\sum _ { i = 1 }

Question 8

Use mathematical induction to prove the following.
-$$1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \ldots + n ( n + 2 ) = \frac { n ( n + 1 ) ( 2 n + 7 ) } { 6 }$$

Accepted Answer

Answers may vary. One possibil

Question 9

Find the nth, or general, term.
-$$\frac { 1 } { 7 } , \frac { 1 } { 49 } , \frac { 1 } { 343 } , \ldots$$

Accepted Answer

A)  $7 ^ { n - 1 }$ 
B)  $\frac { 1 } { 7 ^ { n - 1 } + 6 }$ 
C)  $\frac { 1 } { 7 + 42 ( n - 1 ) }$ 
D)  $\frac { 1 } { 7 ^ { n } }$ 
A)  $7 ^ { n - 1 }$ 
B)  $\frac { 1 } { 7 ^ { n - 1 } + 6 }$ 
C)  $\frac { 1 } { 7 + 42 ( n - 1 ) }$ 
D)  $\frac { 1 } { 7 ^ { n } }$

Question 10

Solve.
-How many 4-letter codes can be formed with the letters A, B, C, D, E, F, G, H with repetition?

Accepted Answer

A) 4096 
B) 336 
C) 240 
D) 56 
A) 4096 
B) 336 
C) 240 
D) 56

Question 11

Find the indicated term of the binomial expansion.
-5 th term; $( 2 x + 5 ) ^ { 5 }$

Accepted Answer

A)  $2500 x ^ { 2 }$ 
B)  $6250 x$ 
C)  15,625 
D)  $1250 x$ 
A)  $2500 x ^ { 2 }$ 
B)  $6250 x$ 
C)  15,625 
D)  $1250 x$

Question 12

Evaluate.
-${ } _ { 20 } \mathrm { P } _ { 4 }$

Accepted Answer

A)  6840 
B)  $2.43 \times 10^{18}$ 
C)  116,280 
D)  $1,860,480$ 
A)  6840 
B)  $2.43 \times 10^{18}$ 
C)  116,280 
D)  $1,860,480$

Find the indicated quantity. - $a _ { 31 }$ , when $\mathrm { a } _ { 1 } = 1$ and $\mathrm { d } = - \frac { 7 } { 6 }$

Find the nth, or general, term. - $25,5,1 , \ldots$

Expand. - $\left( \frac { 5 } { x } - 3 x \right) ^ { 4 }$

Use mathematical induction to prove the following. - $\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \ldots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$

Use mathematical induction to prove the following. - $6 + 12 + 18 + \ldots + 6 n = 3 n ( n + 1 )$

Find the sum, if it exists. - $8 - 9 + \frac { 81 } { 8 } - \frac { 729 } { 64 } + \ldots$

Use mathematical induction to prove the following. - $\sum _ { i = 1 } ^ { n } ( 5 i + 8 ) = \frac { n ( 5 n + 21 ) } { 2 }$

Use mathematical induction to prove the following. - $1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \ldots + n ( n + 2 ) = \frac { n ( n + 1 ) ( 2 n + 7 ) } { 6 }$

Find the nth, or general, term. - $\frac { 1 } { 7 } , \frac { 1 } { 49 } , \frac { 1 } { 343 } , \ldots$

Solve. -How many 4-letter codes can be formed with the letters A, B, C, D, E, F, G, H with repetition?

Find the indicated term of the binomial expansion. -5 th term; $( 2 x + 5 ) ^ { 5 }$

Evaluate. - ${ } _ { 20 } \mathrm { P } _ { 4 }$

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Exam 8: Sequences, Series, and Combinatorics

Find the indicated quantity. - a31a _ { 31 }a31​ , when a1=1\mathrm { a } _ { 1 } = 1a1​=1 and d=−76\mathrm { d } = - \frac { 7 } { 6 }d=−67​

Find the nth, or general, term. - 25,5,1,…25,5,1 , \ldots25,5,1,…

Expand. - (5x−3x)4\left( \frac { 5 } { x } - 3 x \right) ^ { 4 }(x5​−3x)4

Use mathematical induction to prove the following. - (1−12)(1−13)…(1−1n+1)=1n+1\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \ldots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }(1−21​)(1−31​)…(1−n+11​)=n+11​

Use mathematical induction to prove the following. - 6+12+18+…+6n=3n(n+1)6 + 12 + 18 + \ldots + 6 n = 3 n ( n + 1 )6+12+18+…+6n=3n(n+1)

Find the sum, if it exists. - 8−9+818−72964+…8 - 9 + \frac { 81 } { 8 } - \frac { 729 } { 64 } + \ldots8−9+881​−64729​+…

Use mathematical induction to prove the following. - ∑i=1n(5i+8)=n(5n+21)2\sum _ { i = 1 } ^ { n } ( 5 i + 8 ) = \frac { n ( 5 n + 21 ) } { 2 }∑i=1n​(5i+8)=2n(5n+21)​

Use mathematical induction to prove the following. - 1⋅3+2⋅4+3⋅5+…+n(n+2)=n(n+1)(2n+7)61 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \ldots + n ( n + 2 ) = \frac { n ( n + 1 ) ( 2 n + 7 ) } { 6 }1⋅3+2⋅4+3⋅5+…+n(n+2)=6n(n+1)(2n+7)​

Find the nth, or general, term. - 17,149,1343,…\frac { 1 } { 7 } , \frac { 1 } { 49 } , \frac { 1 } { 343 } , \ldots71​,491​,3431​,…