Question

Determine first five terms of sequence?

Determine the first five terms of the sequence. See image for the problem.

a1=2 and a2=3. What are the other three numbers in the first five terms of the sequence?

Answer 1

`2, 3, 22, 103, 522`

Bonus: `a_n = -7/6(-1)^n+1/6(5)^n`

Explanation:

Given:

`{ (a_1 = 2), (a_2 = 3), (a_n = 4a_(n-1)+5a_(n-2)) :}`

The first `5` terms are:

`a_1 = 2`

`a_2 = 3`

`a_3 = 4a_2+5a_1 = 4(color(blue)(3))+5(color(blue)(2)) = 12+10 = 22`

`a_4 = 4a_3+5a_2 = 4(color(blue)(22))+5(color(blue)(3)) = 88+15 = 103`

`a_5 = 4a_4+5a_3 = 4(color(blue)(103))+5(color(blue)(22)) = 412+110 = 522`

Bonus - What is a formula for a general term of this sequence?

Focusing on the recurrence rule step:

`a_n = 4a_(n-1)+5a_(n-2)`

is there a geometric sequence which obeys this rule?

If so, then its common ratio `r` must satisfy:

`r^2-4r-5 = 0`

That is:

`(r+1)(r-5) = 0`

So `" "r = -1" "` or `" "r = 5`

Hence the general term of the given sequence must be expressible in the form:

`a_n = A(-1)^n + B(5)^n`

Using our values for `a_1` and `a_2`, we find:

`2 = a_1 = A(-1)^1+B(5)^1 = -A+5B`

`3 = a_2 = A(-1)^2+B(5)^2 = A+25B`

Adding these two equations, we find `30B=5` and hence `B=1/6`

Then using this value for `B` in the first equation, we find:

`A=5B-2 = 5/6-2 = -7/6`

So:

`a_n = -7/6(-1)^n+1/6(5)^n`