HELP ME PLEASEE Given that 4, 5x and 16 are the first three terms of geometric progression

Find:
(1)
the possible values of x,

(2)
the sum of the first ten terms if all the terms are positive.
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Answers:

x = -8/5 or x = 8/5

Sum of the first ten terms where all terms are positive = 4092

Explanation:

r = common ratio

The first three terms are: 4, 4r, 4r^2

We're given that the sequence is: 4, 5x, 16

Therefore, we have these two equations

Solve the second equation for r and you should find that r = -2 or r = 2 are the only possible solutions. If r = -2, then 5x = 4r solves to x = -8/5. If r = 2, then 5x = 4r solves to x = 8/5.

To find the sum of the first n terms, we use this geometric series formula

Sn = a*(1 - r^n)/(1 - r)

We have

So,

Sn = a*(1 - r^n)/(1 - r)

S10 = 4*(1 - 2^10)/(1 - 2)

S10 = 4*(1 - 1024)/(-1)

S10 = 4*(-1023)/(-1)

S10 = 4092