Your working looks good, to finish it,
Solve for where
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.
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.
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)?