Exam 12: Sequences and Series

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

A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month, and the farmer harvests 200 catfish per month. The catfish population $P _ { n }$ after n months is given recursively by $P _ { 0 } = 5000$ and $P _ { n } = 1.08 P _ { n - 1 } - 200$ . How many fish are in the pond after 5 months?

Determine the common difference, the fifth, $n ^ { \text {th } }$ , and $100 ^ { \text {th } }$ terms of the arithmetic sequence $1,2,3,4 , \ldots$ .

Determine whether the sequence $1,5,25,125 , \ldots$ is geometric. If it is geometric, find the common ratio.

Find the $n ^ { \text {th } }$ term of the arithmetic sequence given $a = 4$ and $d = - 3$ .

Write the sum $\sum _ { k = 1 } ^ { 5 } \left( \frac { \sqrt { k } } { k + 1 } \right) ^ { k - 1 }$ without using sigma notation.

Use mathematical induction to prove that the formula $1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \cdots + n ( n + 2 ) = \frac { n ( n + 1 ) ( 2 n + 7 ) } { 6 }$ is true for all natural numbers $n$ .

Use Pascal's Triangle to expand the expression $\left( 2 x ^ { 2 } - y ^ { 2 } \right) ^ { 5 }$ .

Given the partial sum of an arithmetic sequence $2.1 + 2.7 + 3.3 + \cdots + 13.5$ . Find its sum.

Find the partial sum $S _ { 7 }$ of the geometric sequence with $a = 3$ , $r = 2$ .

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

Find the term of the sequence whose first several terms are $3$ , $9$ , $27$ , $81$ , ... .

Determine the common ratio, the fifth, and the $n ^ { t h }$ terms of the geometric sequence $5 , \frac { 15 } { 4 } , \frac { 45 } { 16 } , \frac { 135 } { 64 } , \cdots$

A city was incorporated in 2004 with a population of 25,000. 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 $P _ { n } = 25000 ( 1.02 ) ^ { n }$ (a) Find the first 5 terms of the sequence. (b) Find the population in 2014.

Find the fourth term and the $n ^ { t h }$ term of the geometric sequence given $a = 7$ and $r = \frac { 1 } { 7 }$ .

Find the first four terms and the term of the sequence $a _ { \mathrm { n } } = 7$ .

A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month, and the farmer harvests 200 catfish per month. The catfish population $P _ { n }$ after n months is given recursively by $P _ { 0 } = 5000$ and $P _ { n } = 1.08 P _ { n - 1 } - 200$ . How many fish are in the pond after 6 months?

A couple can afford to pay $825 per month toward buying a house. If the mortgage rate is 6.25% and they intend to secure a 15year mortgage, how much can they borrow? How much can they borrow if they intend on securing a 30 year mortgage?

A culture initially has 6000 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.

Determine the common ratio, the fifth, and the $n ^ { t h }$ terms of the geometric sequence $2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots$ .