# maths

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If told that the 3rd term of a GP is 36 and the 8th term is 8748. find the first term and the common ration.

• maths -

start with 4 with a factor of 3

4 12 36 for the 3rd

• maths -

The n - th term of a geometric progression with initial value a and common ratio r is given by:

an = a * r ^ ( n - 1 )

In this case :

a3 = a * r ^ ( 3 - 1 ) = a * r ^ 2 = 36

a8 = a * r ^ ( 8 - 1 ) = a * r ^ 7 = 8748

So you must solve two equations :

a * r ^ 2 = 36

and

a * r ^ 7 = 8748

a * r ^ 2 = 36 Divide both sides by r ^ 2

a * r ^ 2 / r ^ 2 = 36 / r ^ 2

a = 36 / r ^ 2

a * r ^ 7 = 8748 Divide both sides by r ^ 7

a * r ^ 7 / r ^ 7 = 8748 / r ^ 7

a = 8748 / r ^ 7

a = a

36 / r ^ 2 = 8748 / r ^ 7 Multiply both sides by r ^ 7

36 * r ^ 7 / r ^ 2 = 8748 * r ^ 7 / r ^ 7

36 * r ^ 5 = 8748 Divide both sides by 36

36 * r ^ 5 / 36 = 8748 / 36

r ^ 5 = 243

r = fifth root ( 243 )

r = 3

a = 36 / r ^ 2

a = 36 / 3 ^ 2 = 36 / 9 = 4

OR

a = 8748 / r ^ 7

a = 8748 / 3 ^ 7 = 8748 / 2187 = 4

The first term of a GP:

a = 4

The common ratio:

r = 3

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