Question 1092222: if a15 = 20, a30 = -13 then what is r, S50 (Geometric Sequence)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula for An of a geometric sequence is:
An = A1 * r ^ (n-1)
the formula for Sn of a geometric sequence is:
Sn = A1 * (1 - r ^ n) / (1 - r)
we first need to find A1 and r and then we can find Sn.
you are given that:
A15 = 20
A30 = -13
if you know that An = A1 * r ^ (n-1), and you replace A1 with Am, then it follows that the formula should then read:
An = Am * r ^ (n-m)
therefore, you get:
A30 = A15 * r ^ (30 - 15)
this becomes A30 = A15 * r ^ (15)
since A30 = -13 and A15 = 20, you get -13 = 20 * r ^ (15)
divide both sides of this equation by 20 to get -13 /20 = r ^ (15)
take the 15th root of both sides of this equation to get (-13 / 20) ^ (1/15) = r
solve for r to get r = -.9716896058
now that you know r, you can solve for A1.
the formula of An = A1 * r ^ (n-1) becomes A15 = A1 * r ^ 14 when n = 15.
since A15 = 20 and r = -.9716896058, this formula then becomes 20 = A1 * -.9716896058 ^ 14.
divide both sides of this equation by -.9716896058 ^ 14 and you get 20 /(-.9716896058 ^ 14) = A1.
solve for A1 to get A1 = 29.89814172
now that you know A1 and r, you can solve for S50.
the formula for Sn is Sn = A1 * (1 - r^n) / (1 - r)
when n = 50 and A1 = 29.89814172 and r = -.9716896058, that formula becomes S50 = 29.89814172 * (1 - (-.9716896058) ^ 50) / (1 - (-.9716896058).
solve for S50 to get S50 = 11.55640593
i took the trouble to do the actual n by n calculations for you in excel so you could see the progression.
this is what it looks like.
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