Exam 11: Sequences; Induction; the Binomial Theorem

Solve the problem. -Use the Binomial Theorem to approximate $( 0.98 ) ^ { 6 } = \left( 1 - 2 \left( 10 ^ { - 2 } \right) \right) ^ { 6 }$ to 5 decimal places.

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 4 } ( - k )$

Write out the first five terms of the sequence. - $\left\{ \frac { ( 2 n - 1 ) ! } { n ! } \right\}$

Find the nth term of the geometric sequence. -5, -15, 45, -135, 405 A) $a _ { n } = 5 \cdot ( - 3 ) ^ { n - 1 }$ B) $a _ { n } = 5 \cdot ( - 3 ) n$ C) $a _ { n } = a _ { 1 } - 3 ^ { n }$ D) $a _ { n } = 5 \cdot ( - 3 ) ^ { n }$

Find the indicated term using the given information. - $\mathrm { a } _ { 45 } = - \frac { 147 } { 5 } , \mathrm { a } _ { 13 } = - \frac { 51 } { 5 } ; \mathrm { a } _ { 3 }$

Find the sum of the arithmetic sequence. -{-6n + 2}, n = 36

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. -4, -12, 36, -108, 324

Find the sum of the sequence. - $\sum _ { k = 2 } ^ { 5 } ( - 1 ) ^ { k + 1 } ( k - 9 ) ^ { 2 }$

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. - $\sum _ { k = 1 } ^ { 5 } 3 ( - 4 ) ^ { k }$

The given pattern continues. Write down the nth term of the sequence suggested by the pattern. -0, 2, 6, 12, 20, ... A) $a _ { n } = 4 n - 6$ B) $a _ { n } = 2 ^ { n - 1 } - 1$ C) $a _ { n } = n ^ { 2 } - n$ D) $a _ { n } = 2 n - 2$

The sequence is defined recursively. Write the first four terms. - $a _ { 1 } = 3 \text { and } a _ { n } = 2 a _ { n - 1 } \text { for } n \geq 2$

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $\left( 3 ^ { 5 } \right) ^ { n } = 3 ^ { 5 n }$

Write out the first five terms of the sequence. - $\left\{ \frac { 4 } { n ^ { 2 } } \right\}$

Solve the problem. -Suppose you just received a job offer with a starting salary of $37,000 per year and a guaranteed raise of $1500 per year. How many years will it be before you've made a total (or aggregate) salary of $1,025,000?

Write out the sum. Do not evaluate. - $\sum _ { k = 1 } ^ { 6 } ( 5 k - 3 )$

Solve the problem. -For the geometric sequence 2, 1, $2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots , \text { find } a _ { n }$ . A) $a _ { n } = 2 ^ { 1 - n }$ B) $a _ { n } = \left( \frac { 1 } { 2 } \right) ^ { 1 - n }$ C) $a _ { n } = 2 ^ { n - 2 }$ D) $a _ { n } = \left( \frac { 1 } { 2 } \right) ^ { n - 2 }$

Expand the expression using the Binomial Theorem. - $( w - s ) ^ { 6 }$

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

Find the sum. -2, -4, 8, -16, 32

Determine whether the sequence is arithmetic. -5, -15, 45, -135, 405, ...