Exam 11: Sequences; Induction; the Binomial Theorem

Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $\frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } + \ldots + \frac { 1 } { 2 ^ { n } } = 1 - \frac { 1 } { 2 ^ { n } }$

Find the sum of the arithmetic sequence. -{-6n - 1}, n = 38

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given. - $a = 5 ; r = 3 \pi$

A geometric sequence is given. Find the common ratio and write out the first four terms. - $\left\{ 2 \left( \frac { 1 } { 2 } \right) ^ { n - 1 } \right\}$

Find the indicated coefficient or term. -The coefficient of $x$ in the expansion of $( 7 x + 3 ) ^ { 3 }$

Choose the one alternative that best completes the statement or answers the question. -The 3rd term in the expansion of $( 7 x + 9 ) ^ { 3 }$

Write the word or phrase that best completes each statement or answers the question. -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?

Find the value of the annuity with the following characteristics. -Payments of $1000 are made at the end of each year for 15 years at 9% interest compounded annually

Evaluate the factorial expression. - $\frac { 7 ! } { 5 ! 2 ! }$

Find the indicated term using the given information. - $\mathrm { a } = 2 , \mathrm {~d} = \frac { 7 } { 5 } ; \mathrm { a } 26$

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 16 } ( 2 k + 7 )$

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

Choose the one alternative that best completes the statement or answers the question. - $a _ { 1 } = \sqrt { 6 } ; a _ { n } = \sqrt { 6 a _ { n - 1 } }$

The given pattern continues. Write down the nth term of the sequence suggested by the pattern. - $2,4,8,16,32 , \ldots$

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a1, and common ratio, r. -Find ag when $\mathrm { a } _ { 1 } = 4,000 , \mathrm { r } = \frac { 1 } { 3 }$ .

Choose the one alternative that best completes the statement or answers the question. -The 10 th term in the expansion of $( 3 x - 2 y ) ^{13}$

Solve the problem. -A theater has 20 rows with 23 seats in the first row, 27 in the second row, 31 in the third row, and so forth. How many seats are in the theater?

Find the sum of the arithmetic sequence. - $- 3 + 1 + 5 + 9 + 13 + \ldots + ( 4 n - 7 )$

Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $1 + 3 + 3 ^ { 2 } + \ldots + 3 ^ { n - 1 } = \frac { 3 ^ { n } - 1 } { 2 }$

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