Exam 12: Sequences and Series

A man gets a job with a salary of $50,000 a year. He is promised an $1800 raise each subsequent year. Find his total earnings for a 10-year period.

Which term of the arithmetic sequence $\frac { 1 } { 2 } , 2 , \frac { 7 } { 2 } , \ldots$ is $23$ ?

The seventh term of an arithmetic sequence is $- 16$ and the seventeenth term is $- 66$ . Find the twenty-fourth term.

Find the sum $5 + 25 + 125 + \ldots + 15625$ .

Use mathematical induction to prove that the sum of a finite arithmetic sequence is $\frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$ .

Find the first five terms, the common ratio, and the $n ^ { t h }$ term expressed in standard form of the geometric sequence $a _ { \mathrm { n } } = \frac { ( - 1 ) ^ { n } } { 3 ^ { n - 1 } }$ .

Find the middle term in the expansion of $( x - 2 y ) ^ { 14 }$ .

Which term of the geometric sequence $5,15,45,135 , \ldots$ is $32,805$ ?

The first term of a geometric sequence is $12$ and the second term is $4$ . Find the fifth term.

Dr. Stevens is considering a 30-year mortgage at 4% interest. She can make payments of $3500 a month. What size loan can she afford?

Determine the common ratio, the fifth, and the $n ^ { t h }$ terms of the geometric sequence $7,7 ^ { 7 / 6 } , 7 ^ { 4 / 3 } , 7 ^ { 3 / 2 } , \ldots$

Find the term of the sequence whose first several terms are $3$ , $- \frac { 1 } { 9 }$ , $27$ , $- \frac { 1 } { 81 }$ , ... .

Find the first five terms of the sequence $a _ { n } = 3 \left( a _ { n - 1 } + 1 \right) , \quad a _ { 1 } = 1$ , which is defined recursively.

Find the amount of an annuity that consists of 8 monthly payments of $1200 each into an account that pays $24 \%$ interest per year.

Determine whether the sequence $3,9,15,21 , \ldots$ is arithmetic. If it is arithmetic, find the common difference.

Find the coefficient of $x ^ { 8 } y ^ { 6 }$ in the expansion of $( y - 2 x ) ^ { 14 }$ .

How much must be invested now at 6.5% per year compounded quarterly, to fund an annuity of 15 quarterly payments of $500 each, the first payment being three months after the initial investment?

An arithmetic sequence has first term $a = - 2$ and common difference $d = 4$ . How many terms of this sequence must be added to get $1566 ?$

Find the second term in the expansion of $\left( x ^ { 2 } - \frac { 1 } { x } \right) ^ { 45 }$

Determine the common difference, the fifth, $n ^ { \text {th } }$ , and $100 ^ { \text {th } }$ terms of the arithmetic sequence $31,28,25,22 , \ldots$ .