Guessing Explicit Formula for a Sequence

imdaman

1. The problem statement, all variables and given/known data

Given a recursive sequence, use iteration to guess an explicit formula
a(k) = 5a(k-1) + k and a(1) = 1

2. Relevant equations

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


3. 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 dont 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?