Exam 12: Sequences and Series

Use Pascal's triangle to expand the expression $\left( \frac { 1 } { y } + y \right) ^ { 5 }$ .

Find the first four terms and the term of the sequence $a _ { n } = ( - 1 ) ^ {^{ n} } {\frac { n + 2 } { n }}$ .

How much money should be invested monthly at 18% per year, compounded monthly, in order to have $\$ 2000$ in ten months?

A farmer wants to borrow $50,000 in order to buy a tractor. She wants to repay the loan by semi-annual installments for six years. If the interest rate is 10% per year compounded semi-annually, what is the amount of each payment?

Find the 22^nd term in the expansion of $( x y - 1 ) ^ { 24 }$ .

Find the sum of the first 50 even integers.

Factor using the Binomial Theorem. $a ^ { 3 } + 6 a ^ { 2 } b + 12 a b ^ { 2 } + 8 b ^ { 3 }$

Express the repeating decimal $0.345345345 \ldots$ as a fraction.

How much money must be invested now at 5% per year compounded semi-annually, to fund an annuity of $14$ semi-annual payments of $\$ 100$ each, the first payment being six months after the initial investment?

Factor using the Binomial Theorem. $x ^ { 5 } + 5 x ^ { 4 } y + 10 x ^ { 3 } y ^ { 2 } + 10 x ^ { 2 } y ^ { 3 } + 5 x y ^ { 4 } + y ^ { 5 }$

Find the term of the sequence $\frac { 1 } { 4 } , \frac { - 1 } { 9 } , \frac { 1 } { 16 } , \frac { - 1 } { 25 } , \ldots$ .

Determine the common difference, the fifth, $n ^ { \text {th } }$ , and $100 ^ { \text {th } }$ terms of the arithmetic sequence $- 1,7,15,23 , \ldots$ .

A ball rebounds to one-quarter the height from which it was dropped. Approximate the total vertical distance the ball travels after being dropped from $3$ ft above the ground, until it comes to rest.

Use Pascal's Triangle to expand the expression $\left( \frac { 1 } { y } - \alpha \right) ^ { 4 }$ .

Determine the common difference, the fifth, $n ^ { \text {th } }$ , and $100 ^ { \text {th } }$ terms of the arithmetic sequence $2,4,6 , \ldots$ .

A man gets a job with a salary of $40,000 a year. He is promised an $1800 raise each subsequent year. Find his total earnings for a 10-year period.

A culture initially has 7000 bacteria, and its size increases by 8% every hour. How many bacteria are present at the end of 5 hours? Find a formula for the number of bacteria present after n hours.

Alyson buys a computer by paying $15 a week for three years. If the interest rate is 10.4% per year, compounded weekly, what is the price of the computer?

Find the first four terms and the term of the sequence $a _ { n } = ( - 1 ) ^ { n } n ^ { 2 }$ .

Find the first four terms and the term of the sequence $a _ { \mathrm { n } } = \frac { 2 } { n + 3 }$ .