Guessing Explicit Formula for a Sequence

[imdaman]

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?

 

[mighty2000]

Hi there,

Thank you for sharing your attempt at solving this problem. It seems like you are on the right track with identifying the pattern for the power of 5. However, you are correct that the extra multiplication term is throwing off the pattern. In order to solve this, you can try to factor out a 5 from each term, as you mentioned.

Let's take a look at the first few terms:

a(1) = 1
a(2) = 5a(1) + 2 = 5*1 + 2 = 7
a(3) = 5a(2) + 3 = 5*7 + 3 = 38
a(4) = 5a(3) + 4 = 5*38 + 4 = 194

As you can see, the first term is always 5^(n-1), but the second term is not always (n-1). However, if we factor out a 5, we get:

a(4) = 5*(5*7 + 3) + 4 = 5*5*7 + 5*3 + 4 = 5^2*7 + 5^1*3 + 5^0*4

Now, if we continue this pattern, we can see that the second term is always (n-1), but the first term is changing. Let's try to write this in a general form:

a(n) = 5^(n-1)*(n-1) + b

Where b is a constant term that we need to determine. To find b, we can use the initial condition a(1) = 1:

a(1) = 5^(1-1)*(1-1) + b = 0 + b = 1
b = 1

Therefore, the explicit formula for this recursive sequence is:

a(n) = 5^(n-1)*(n-1) + 1

I hope this helps! Keep up the good work with your problem solving.