Determine first five terms of sequence?

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

1 Answer
Dec 7, 2017

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