What should be the value of x so that the three consecutive terms will be in GP 2x+2, 16 and 64x. Also find the sum of first 10 terms if the starting term is 2x+2.

x = 1 and -2

Sum of the first ten terms = 1,398,100

Step-by-step explanation:

Given the geometric progression 2x+2, 16, 64x...

Since they are in GP, their common ratio is expressed as;

16/2x+2 = 64x/16

16/2(x+1) = 4x

8/x+1 = 4x

Cross multiply

4x(x+1) = 8

4x²+4x = 8

4x²+4x - 8 = 0

divide through by 4

x²+x - 2 = 0

Factorize

x²+2x-x- 2 = 0

x(x+2)-1(x+2) = 0

(x-1)(x+2) = 0

x = 1 and -2

Hence the values of x are 1 and -2

Get the common ratio;

r = 16/2x+2

Since x = 1

r = 16/2(1)+2

r = 16/4

r = 4

Sum of the first n terms is expressed as;

Sn = a(rⁿ-1)/r-1

a = 2x+2

a = 2(1)+2

a = 4

Substitute;

S10 = 4(4^10 - 1)/4 - 1

S10 = 4(1,048,576-1)/3

S10 = 4,194,300/3

S10 = 1,398,100

Hence the sum of the first ten terms is 1,398,100

fist on the left goes with the third on the right. second on the left goes with the fourth on the right. third on the left goes with the second on the right. the fourth on the left goes with first on the right