SOLUTION: 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.

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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