CSE$2500-$ Assignment $7$

$($a$)$ What is a common ratio? What is a common difference? $(2$ pts$)$

$($b$)$ What is the common ratio of the following geometric sequence? $(1$ pts$)27,9,3,1,$dots

$($c$)$ What is $a_3$ in the following sequences? $(3$ pts$)$

A geometric sequence starts with $a_0=\frac{1}{4}.$ The common ratio is $r=2.a_3$ is equal to

An arithmetic sequences starts with $a_1=4.$ The common difference is $d=-3.a_3$ is equal to

A sequence $\{a_n\}$ is defined as follows: $a_0=2,a_1=1,$ and for $n2,a_n=3*a_{n-1}-n*a_{n-2}+$

$a_3$ is equal to

A population of mice increases by $10\%$ every year. Define $g_n$ to be the number of mice after $n$ years.

Write the recurrence relation that describes the sequence $\{g_n\}.(2$ pts$)$

Write the summation expression whose value is equivalent to the sum $4^3+6^3+8^3+$cdots$+{28}^3.(2$

pts$)$

$($a$)$ What is the difference between mathematical induction and strong induction? $(4$ pts$)$

$($b$)$ Define the sequence $\{a_n\}$ as follows:

$a_0=a_1=2$

$a_n=(a_{n-1}{)}^2{a}_{n-2},$ for $n2$

Prove that for all $n0,a_n{2}^{3^n}$ using strong induction. $(6$ pts$)$