Exam 12: Sequences and Series

The first term of the arithmetic sequence is $\frac { 2 } { 3 }$ and the common difference is $\left( - \frac { 2 } { 3 } \right)$ . Which term of this sequence is $- \frac { 20 } { 3 }$ ?

What is the monthly payment on a 50 year mortgage of $500,000 at 12% interest per year, compounded monthly? What is the total amount paid on this loan over the 50 year period?

Find the sum of the infinite geometric series $\frac { 2 } { \sqrt { 2 } } + 1 + \frac { 1 } { - \sqrt { 2 } } + \frac { 1 } { 2 } + \cdots$ .

Determine whether the sequence $1,4,9,16 , \ldots$ is geometric. If it is geometric, find the common ratio.

Determine whether the sequence $1,4,7,10 , \ldots$ is arithmetic. If it is arithmetic, find the common difference.

Factor using the Binomial Theorem. $x ^ { 4 } - 12 x ^ { 3 } y + 54 x ^ { 2 } y ^ { 2 } - 108 x y ^ { 3 } + 81 y ^ { 4 }$

Mike buys a stereo by paying $50 a month for two years. If the interest rate is 12% per year, compounded monthly, what is the price of the stereo?

Simplify using the Binomial Theorem. $\frac { ( x - h ) ^ { 4 } - x ^ { 4 } } { h }$

What is the monthly payment on a 20 year mortgage of $85,000 at 9% interest per year, compounded monthly? What is the total amount paid on this loan over the 20 year period?

Find the term of the sequence whose first several terms are $\frac { 1 } { 2 }$ , $4$ , $\frac { 1 } { 8 }$ , $16$ , ... .

Determine the common difference, the fifth, $n ^ { \text {th } }$ , and $100 ^ { \text {th } }$ terms of the arithmetic sequence $7,10,13,16 , \ldots$ .

Find the second term in the expansion of $\left( x ^ { 2 } - \frac { 1 } { x } \right) ^ { 50 }$

Find the first six partial sums $S _ { 1 }$ , $S _ { 2 }$ , $S _ { 3 }$ , $S _ { 4 }$ , $S _ { 5 }$ , and $S _ { 6 }$ of the sequence $1,8,27,64 , \ldots$ . .

Use mathematical induction to prove that the sum of a partial geometric sequence is $\frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$ .

Find the amount of an annuity that consists of 15 annual payments of $200 each into an account that pays $12 \%$ interest per year.

Find the amount of an annuity that consists of 30 annual payments of $500 each into an account that pays $7 \%$ interest per year.

Find the first five terms, the common ratio, and the $n ^ { t h }$ term expressed in standard form of the geometric sequence $a _ { n } = ( 3 ) 2 ^ { n }$ .

Given that the $50 \text { th }$ term of an arithmetic sequence is $259$ and the common difference is $5$ , find the first three terms.

Use Pascal's triangle to expand the expression $\left( 3 - \frac { x } { 3 } \right) ^ { 4 }$ .

Determine the common difference, the fifth, $n ^ { \text {th } }$ , and $100 ^ { \text {th } }$ terms of the arithmetic sequence $11,11.2,11.4,11.6 , \ldots$ .