Exam 11: Sequences; Induction; the Binomial Theorem

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence. -7th term is 45; 16th term is 117 A) $a _ { 1 } = - 3 , d = 8 , a _ { n } = a _ { n - 1 } - 8$ B) $a _ { 1 } = - 3 , d = 8 , a _ { n } = a _ { n - 1 } + 8$ C) $a _ { 1 } = - 11 , d = 8 , a _ { n } = a _ { n - 1 } - 8$ D) $a _ { 1 } = - 11 , d = 8 , a _ { n } = a _ { n - 1 } + 8$

Determine whether the sequence is geometric. -3, 6, 12, 24, 48, ...

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $4 - 1 + \frac { 1 } { 4 } - \cdots$

Expand the expression using the Binomial Theorem. - $( g - 2 h ) ^ { 3 }$

Solve the problem. -Write 0.272727... as a fraction. (Find the sum of the repeating decimal.) A) $\frac { 3 } { 11 }$ B) $\frac { 6 } { 11 }$ C) $\frac { 3 } { 10 }$ D) $\frac { 1 } { 37 }$

Solve the problem. -Suppose that certain bacteria can double their size and divide every 30 minutes. Write a recursive sequence that describes this growth where each value of n represents a 30-minute interval. Let $\mathrm { a } _ { 1 } = 490$ represent the initial Number of bacteria present. A) $a _ { n } = 2 a _ { 1 }$ for all values of $n$ . B) $a _ { 1 } = 490 ; a _ { n } = 2 a _ { n + 1 }$ for $n > 1$ . C) $a _ { 1 } = 490 ; \quad a _ { n } = 2 a _ { n - 1 }$ for $n > 1 .$ D) $a _ { 1 } = 490 ; \quad a _ { n } = \frac { 1 } { 2 } a _ { n - 1 }$ for $n > 1$ .

The sequence is defined recursively. Write the first four terms. - $a _ { 1 } = \sqrt { 6 } ; a _ { n } = \sqrt { 6 a _ { n - 1 } }$

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$

Find the sum of the arithmetic sequence. -6 + 12 + 18 + ... + 876

Find the sum of the arithmetic sequence. -{2n - 2}, n = 43

Determine whether the sequence is arithmetic. -4, 12, 36, 108, 972, ...

Expand the expression using the Binomial Theorem. - $( 2 x + 1 ) ^ { 5 }$

Find the indicated coefficient or term. -The coefficient of $\frac { 1 } { x }$ in the expansion of $\left( 2 x + \frac { 1 } { x } \right) ^ { 3 }$

Solve the problem. -A new piece of equipment cost a company $54,000. Each year, for tax purposes, the company depreciates the value by 25%.What value should the company give the equipment after 8 years?

Determine whether the given sequence is arithmetic, geometric, or neither. If arithmetic, find the common difference. If geometric, find the common ratio. - $\left\{ \left( \frac { 4 } { 3 } \right) ^ { n } \right\}$

Find the indicated term of the sequence. -The twentieth term of the arithmetic sequence 11 , 7 , 3 , ...

Find the indicated coefficient or term. -The 10 th term in the expansion of $( 3 x - 2 y ) ^ { 13 }$

Solve the problem. -A pendulum bob swings through an arc 40 inches long on its first swing. Each swing thereafter, it swings only 63% as far as on the previous swing. What is the length of the arc after 9 swings? Round your answer to two Decimal places, if necessary.

Determine whether the sequence is arithmetic. -2, 4, 6, 10, 12, ...

Solve the problem. -Find the 10th term of the geometric sequence $\frac { 1 } { 3 } , 1,3 , \ldots$