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Question 898852: Please help me solve this question: In the gp, the sum of the 2nd and 3rd terms is 9. The 7th is 8 times the 4th. Find the first term, the ratio and the 5th term.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you are given that:
the sum of the 2d and 3d terms is 9.
the 7th term is 8 times the 4th.
in a geometric progression, the formula for An is:
An = A1 * r^(n-1)
This means that A7 = A1 * r^6
This means that A4 = A1 * r^3
if A7 is 8 * A4, this means that:
A1 * r^6 = 8 * A1 * r^3
divide both sides of this equation by A1 * r^3 and you get:
(A1 * r^6) / (A1 * r^3) = 3
simplify to get:
r^3 = 8 which results in r = 2.
you know that A2 = A1 * r
you also know that A3 = A1 * r^2
since r = 2, this becomes:
A2 = A1 * 2
A3 = A1 * 4
the sum of A2 and A3 is equal to 6 * A1 which is equal to 9
this means that A1 must be equal to 3/2
since A1 is equal to 3/2 and r is equal to 2, this means that A5 is equal to A1 * r^4 which is equal to 3/2 * 2^4) which is equal to 3/2 * 16 which is equal to 24.
the progression from A1 to A7 is as follows:
A1 = 3/2
A2 = 3/2 * 2 = 3
A3 = 3/2 * 4 = 6
A4 = 3/2 * 8 = 12
A5 = 3/2 * 16 = 24
A6 = 3/2 * 32 = 48
A7 = 3/2 * 64 = 96
the sum of the second and third term is equal to 9 because 3 + 6 = 9
the 7th term is 8 times the 4th term because 8 * 12 = 96
your solution is:
first term is equal to 3/2
ratio is equal to 2
fifth term is equal to 24.
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