Finding the sum of a geometric series... help please!

Find the common ratio of a geometric series if the sum is equal to 0.9 and the first term is 0.5.

the formula is s=a/(1-r) where a=first term and s=sum of the series.

0.9=0.5/(1-r)
0.9(1-r)=0.5
0.9-0.9r=0.5
-0.9r=-0.4
r=4/9. So the common ratio is 4/9.

the second term must be .4 so the ratio between the values in the series is 4.:.5 or .4:.8 =]

to find the common ratio or "per term ratio" of any series, you must deduct the value of one term from the sum of the series and write a simplified ratio between one term of the series and another.

A geometric sum is:

S = ar + ar^2 + ar^3 + ar^4 + ... + ar^n

where ar is the first term and r is the common ratio

Assuming that this is an infinite series, n will go to infinity (note that I said it will GO TO infinity, it will not actually be infinity, since infinity is a concept, not a value)

Multiply both sides of the sum by the common ratio of r

Sr = r * (ar + ar^2 + ... + ar^(n - 1) + ar^n)
Sr = ar^2 + ar^3 + ar^4 + ... + ar^n + ar^(n + 1)

Subtract S from Sr

Sr - S = ar^2 + ar^3 + ar^4 + ... + ar^n + ar^(n + 1) - (ar + ar^2 + ar^3 + ... + ar^n)
Sr - S = ar^2 - ar^2 + ar^3 - ar^3 + ar^4 - ar^4 + ... + ar^n - ar^n + ar^(n + 1) - ar
Sr - S = ar^(n + 1) - ar
S * (r - 1) = ar * (r^n - 1)
S = a * r * (r^n - 1) / (r - 1)

Now, since the sum converges, we know that r is between -1 and 1, which means that as n goes to infinity, r^n goes to 0. Also, ar is the first term and we know that to be 0.5. The rest is plug and play

S = 0.9
ar = 0.5
r^inf = 0

0.9 = 0.5 * (0 - 1) / (r - 1)

Solve for r

0.9 * (r - 1) = 0.5 * (0 - 1)
9 * (r - 1) = 5 * (-1)
9r - 9 = -5
9r = 4
r = 4/9

There you go.