- #1

imdaman

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## Homework Statement

Given a recursive sequence, use iteration to guess an explicit formula

a(k) = 5a(k-1) + k and a(1) = 1

## Homework Equations

Sum of Geometric Sequence : (r^(n+1)-1)/(r-1)

## The Attempt at a Solution

y(2) = 5*1 + 2

y(3) = 5(5+2) + 3 = 5^2 5*2 + 3

y(4) = 5(5^2 5*2 + 3) + 4 = 5^3 + 5^2*2 + 5*3 + 4

y(5) = 5(5^3 + 5^2*2 + 5*3 + 4) + 5 = 5^4 + 5^3*2 + 5^2*3 + 5*4 + 5

I see the pattern for the power is k-1 and decreases by 1 but the extra multiplication that increases by 1 is throwing me off.

Should I factor out a 5? I don't really see a pattern though.

I'm not sure if I'm going anywhere with this :

y(5) = 5^(n-1)*(n-4) + 5^(n-2)*(n-3) + 5^(n-3)*(n-2) + 5^(n-4)*(n-1) = 5^(n-1)*(n-(n-1)) + 5^(n-2)*(n-(n-2)) + 5^(n-3)*(n-(n-3)) + 5^(n-4)*(n-(n-4))

Any suggestions?

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