Geometric Series

[PixiePink]

For this equation I'm supposed to find the first term and the sum of the geometric sequence.
an = 24, r = 2/3, n = 4

For this one I need to find the first and last term of the sequence given the information below.
Sn = 1022, r =2, n = 9

I know that the formulas for geometric series are:
Sn=(a1-a1r^n)/1-r & Sn=(a1-ran)/1-r

Im not sure how to solve these though. My book doesn't show how to do these probelms.

 

[stapel]

1) Don't over-think this. You have the fourth term, a₄ = 24, and the common ratio, r = 2/3. But you also know that a₄ = a₁r³ = a₁(2/3)³ = a₁(8/27). So 24 = (8/27)a₁. Solve for the value of a₁.

2) Does "S_n" stand for the n-th partial sum, or for the n-th term?

Eliz.

 

[PixiePink]

Sn is the sum of the first n terms.

Ok so for the first one a1=-81 & sn=-195

but how would I do the second problem?

 

[stapel]

I'd plug the known information into that first summation formula, and solve for a₁. Then back-solve for a₉.

Eliz.

 

[PixiePink]

If you put the information into the first summation you get:
1022=a1-a12^9/1-2
2^9=512 but would you multiply the 512 by each a1 so you get
1022=512a1-512a1/-1 and solve?

 

[stapel]

Why did you convert "a₁ - 512a₁" into "512a₁ - 512a₁"? That would give you zero, which obviously won't work. Just use the formula as they gave it to you.

Eliz.

 

[PixiePink]

Well thats what I don't understand. You get 512 from 2^9 but what do you do with the a1-a1. Would I multiply the 512 by them make them a1^2?

 

[stapel]

You don't have (a₁ - a₁)rⁿ; you have a₁ - (a₁rⁿ). Order of operations: multiply before subtracting.

Eliz.