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)
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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
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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
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the nth term of a geometric progression is
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x(n) = a * r^(n-1)
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we are given
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1) S(4) = 200 = a * ((1 - r^4) / (1 - r))
2) x(5) = 405 = a * r^(5-1) = a * r^4
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we have two equations in two unknowns
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solve equation 2 for a
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a = 405 / r^4
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substitute for a in equation 1
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200 = (405 / r^4) * ((1 - r^4) / (1 - r))
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200 = (405 - 405r^4 / (r^4 - r^5)
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200r^4 - 200r^5 = 405 - 405r^4
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3) 200r^5 - 605r^4 +405 = 0
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a graph of equation 3 shows us that there are three solutions for r (1, 3, and a negative value approximately -0.9)
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we reject r = 1 since that gives us a sum of 0
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by equation 2, a = 5
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to recap we have a = 5 and r = 3
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now we can solve the problem
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16400 = 5 * (1 - 3^n) / (1 - 3)
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divide both sides of = by 5
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3280 = (1 - 3^n) / (-2)
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1 - 3^n = -6560
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-3^n = -6561
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3^n = 6561
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use definition of logarithm
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n = log(3) 6561
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n = 8
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The sum of the first 8 terms equals 16400
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to check this we calculate
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5 * (1 - 3^8) / (1 - 3) =
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5 * -6560 / -2 = 16400
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