## obtain the value of X for which (X+1),(X-5),(X-2) is a geometric progression.hence find the sum of the first 12 terms of the progression.​

Question

obtain the value of X for which (X+1),(X-5),(X-2) is a geometric progression.hence find the sum of the first 12 terms of the progression.​

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6 months 2021-07-17T11:42:24+00:00 1 Answers 9 views 0

## Answers ( )

1. If x + 1, x – 5, and x – 2 are in a geometric progression, then there is some constant r for which

x – 5 = r (x + 1)

==>   r = (x – 5) / (x + 1)

and

x – 2 = r (x – 5)

==>   r = (x – 2) / (x – 5)

Then

(x – 5) / (x + 1) = (x – 2) / (x – 5)

Solve for x :

(x – 5)² = (x – 2) (x + 1)

x ² – 10x + 25 = x ² – x – 2

-9x = -27

x = 3

It follows that the ratio between terms is

r = (3 – 5) / (3 + 1) = -2/4 = -1/2

Now, assuming x + 1 = 4 is the first term of the G.P., the n-th term a(n) is given by

a(n) = 4 (-1/2)ⁿ⁻¹

The sum of the first 12 terms – denoted here by S – is then

S = 4 (-1/2)⁰ + 4 (-1/2)¹ + 4 (-1/2)² + … + 4 (-1/2)¹¹

Solve for S :

S = 4 [(-1/2)⁰ + (-1/2)¹ + (-1/2)² + … + (-1/2)¹¹]

(-1/2) S = 4 [(-1/2)¹ + (-1/2)² + (-1/2)³ + … + (-1/2)¹²]

==>   S – (-1/2) S = 4 [(-1/2)⁰ – (-1/2)¹²]

==>   3/2 S = 4 (1 – 1/4096)

==>   S = 8/3 (1 – 1/4096)

==>   S = 1365/512