Exam 12: Sequences and Series

Find the amount of an annuity that consists of 52 weekly payments of $10 each into an account that pays 3% interest per year.

Find the partial sum $S _ { 5 }$ of the geometric sequence with $a _ { 2 } = 12$ , $a _ { 5 } = 324$ .

Simplify using the Binomial Theorem. $\frac { ( x + h ) ^ { 5 } - x ^ { 5 } } { h }$

Given that the $5 \text { th }$ term of an arithmetic sequence is $30$ and the $7 \text { th }$ term is $44$ , find the first term.

Find the first three terms in the expansion of $\left( x + \frac { 1 } { 2 x } \right) ^ { 40 }$

Write the sum $- 2 x + 4 x ^ { 2 } - 6 x ^ { 3 } + 8 x ^ { 4 } - 10 x ^ { 5 } + \cdots - 98 x ^ { 49 }$ using sigma notation.

Determine whether the sequence $- 1,1 , - 1,1 , \ldots$ is geometric. If it is geometric, find the common ratio.

Determine the common ratio, the fifth, and the $n ^ { t h }$ terms of the geometric sequence. $9,9 \sqrt { 3 } , 27,27 \sqrt { 3 } , \ldots$

Dr. Stevens is considering a 30-year mortgage at 6% interest. She can make payments of $2500 a month. What size loan can she afford?

A large outdoor amphitheater has $99$ rows of seats with $20$ seats in the first row, $24$ seats in the second, $28$ in the third, and so on. How many seats are in row $47 ?$ How many seats are there in the theater? How many seats total are there in rows $32$ through $64 ?$

How much money must be invested now at 16% per year compounded quarterly, to fund an annuity of $20$ quarterly payments of $\$ 1000$ each, the first payment being three months after the initial investment?

Determine whether the sequence $0,1,0,3,0,5 \ldots$ is arithmetic. If it is arithmetic, find the common difference.

Express the repeating decimal $3.1 \overline { 68 }$ as a fraction.

Find the first four partial sums and the partial sum of the sequence $a _ { n } = \ln ( ( n + 1 ) / ( n + 2 ) )$ .

Factor using the Binomial Theorem. $x ^ { 4 } - 4 x ^ { 3 } y ^ { 2 } + 6 x ^ { 2 } y ^ { 4 } - 4 x y ^ { 6 } + y ^ { 8 }$

Kristi decides to buy a new car for $25,000 and make payments of $525 per month for 6 years. What interest rate is she paying on the loan?

Find the sum of the infinite geometric series $\frac { 1 } { 3 } + \frac { 2 } {9 } + \frac { 4 } { 27 } + \frac { 8 } { 81 } + \cdots$ .

How much money should be invested monthly at 6% per year, compounded monthly, in order to have $\$ 10,000$ in two years?

Use Pascal's triangle to expand the expression $( 1 - y ) ^ { 4 }$ .

A city was incorporated in 2004 with a population of 45,000. It is expected that the population will increase at a rate of 2% per year. The population n years after 2004 is given by the sequence $P _ { n } = 45000 ( 1.02 ) ^ { n }$ (a) Find the first 5 terms of the sequence. (b) Find the population in 2014.