Thread: Geometric Sequences and Series

Q. Find Sn, the sum to n terms, of the geometric series 2 + 2/ 3 + 2/ 3^2 +…
If Sn = 242/ 81, find the value of n.

Eq. Sn = a (1 - r^n) / (1 - r) … (r < 1).

Attempt:
From the series above, r = (2/ 3) / 2 => 1/ 3
Therefore, a = 2 and r = 1/ 3. Sub these into the Sn equation…
2 (1 - (1/ 3)^n) / (1 - 1/3) =
2 (1 - (1/ 3)^n) / (2 / 3)

Answer:
I’m stuck at this point, as the answer in the text book indicates that:

Sn = 3 - (1/ (3^(n-1)))

I am uncertain of the procedure they have applied to convert n into n - 1. Can anyone help? Thank you.

Your working looks good, to finish it,

Solve for where

Originally Posted by GrigOrig99

Q. Find Sn, the sum to n terms, of the geometric series 2 + 2/ 3 + 2/ 3^2 +…
If Sn = 242/ 81, find the value of n.

Eq. Sn = a (1 - r^n) / (1 - r) … (r < 1).
...

Answer:
I’m stuck at this point, as the answer in the text book indicates that:

Sn = 3 - (1/ (3^(n-1)))

I am uncertain of the procedure they have applied to convert n into n - 1. Can anyone help? Thank you.

The 1st term in the series is (2)(1/3)^0 . It has an exponent of 0.

The 2nd is (2)(1/3)^1. It has an exponent of (2-1).

The n-th term is 2)(1/3)^(n-1). It has an exponent of (n-1).

Ok, so following on from:
2 (1 - (1/ 3)^n) / (2 / 3)

I have done the following:

= (2(1) - 2(1/ 3)^n) / (2/ 3)
= (2 - (2/3)^n) / (2/ 3)
= (2(3/ 2) - (3/ 2)((2/ 3)^n))
= (6/ 2 - (6/ 6)^n)
= 3 - 1^(n-1)

Recall that the text book answer is Sn = 3 - (1/ (3^(n-1))).

I guess I'm closer to the mark now, but I still don't have it entirely correct. Can anyone see where I've gone wrong? Thanks again.

Originally Posted by GrigOrig99

Ok, so following on from:
2 (1 - (1/ 3)^n) / (2 / 3)

I have done the following:

= (2(1) - 2(1/ 3)^n) / (2/ 3)
You're good to the above line.

= (2 - (2/3)^n) / (2/ 3)
(2)(1/3)^n ≠ (2/3)^n)

= (2(3/ 2) - (3/ 2)((2/ 3)^n))
= (6/ 2 - (6/ 6)^n)
= 3 - 1^(n-1)

Recall that the text book answer is Sn = 3 - (1/ (3^(n-1))).

I guess I'm closer to the mark now, but I still don't have it entirely correct. Can anyone see where I've gone wrong? Thanks again.

Notice that

Ok, I'm almost clear on this. So, is it correct to say that for:

(2.3 (1 - (1/3)^n)/ 2

we do 'not' actually multiply a number (1/3^n, in this case) by another (i.e., 2.3 or 6) when it is expressed to the power of an unknown (n)?

Yes, but we can mulitply them if the power's base is the same as the constant. For example: . (I don't know if this answers your question).

Have you found the value of n now? ...

Great, thanks for clearing that up. n should be 5, then.

Thanks again to everyone who helped.

Right! n=5