SOLUTION: how many terms should be added to make the sum 16400 of a gp having the sum of first four terms 200 and fifth term 405?(p.s 405 is not sum of fifth term..its just fifth term)

Question 1064939: how many terms should be added to make the sum 16400 of a gp having the sum of first four terms 200 and fifth term 405?(p.s 405 is not sum of fifth term..its just fifth term)
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
the sum of n terms of a geometric progression is
:
S(n) = a * ( (1 - r^n) / (1 - r) ), where S(n) is the sum of n terms, a is the first term, r is the common ratio
:
the nth term of a geometric progression is
:
x(n) = a * r^(n-1)
:
we are given
:
1) S(4) = 200 = a * ((1 - r^4) / (1 - r))
2) x(5) = 405 = a * r^(5-1) = a * r^4
:
we have two equations in two unknowns
:
solve equation 2 for a
:
a = 405 / r^4
:
substitute for a in equation 1
:
200 = (405 / r^4) * ((1 - r^4) / (1 - r))
:
200 = (405 - 405r^4 / (r^4 - r^5)
:
200r^4 - 200r^5 = 405 - 405r^4
:
3) 200r^5 - 605r^4 +405 = 0
:
a graph of equation 3 shows us that there are three solutions for r (1, 3, and a negative value approximately -0.9)
:

:
we reject r = 1 since that gives us a sum of 0
:
by equation 2, a = 5
:
to recap we have a = 5 and r = 3
:
now we can solve the problem
:
16400 = 5 * (1 - 3^n) / (1 - 3)
:
divide both sides of = by 5
:
3280 = (1 - 3^n) / (-2)
:
1 - 3^n = -6560
:
-3^n = -6561
:
3^n = 6561
:
use definition of logarithm
:
n = log(3) 6561
:
n = 8
:
*****************************************
The sum of the first 8 terms equals 16400
:
to check this we calculate
:
5 * (1 - 3^8) / (1 - 3) =
:
5 * -6560 / -2 = 16400
******************************************
:

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