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EastWind [94]
2 years ago
15

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 pro

gression.​
Mathematics
1 answer:
Viktor [21]2 years ago
5 0

If <em>x</em> + 1, <em>x</em> - 5, and <em>x</em> - 2 are in a geometric progression, then there is some constant <em>r</em> for which

<em>x</em> - 5 = <em>r</em> (<em>x</em> + 1)

==>   <em>r</em> = (<em>x</em> - 5) / (<em>x</em> + 1)

and

<em>x</em> - 2 = <em>r</em> (<em>x</em> - 5)

==>   <em>r</em> = (<em>x</em> - 2) / (<em>x</em> - 5)

Then

(<em>x</em> - 5) / (<em>x</em> + 1) = (<em>x</em> - 2) / (<em>x</em> - 5)

Solve for <em>x</em> :

(<em>x</em> - 5)² = (<em>x</em> - 2) (<em>x</em> + 1)

<em>x</em> ² - 10<em>x</em> + 25 = <em>x</em> ² - <em>x</em> - 2

-9<em>x</em> = -27

<em>x</em> = 3

It follows that the ratio between terms is

<em>r</em> = (3 - 5) / (3 + 1) = -2/4 = -1/2

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

<em>a(n)</em> = 4 (-1/2)ⁿ⁻¹

The sum of the first 12 terms - denoted here by <em>S</em> - is then

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

Solve for <em>S</em> :

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

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

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

==>   3/2 <em>S</em> = 4 (1 - 1/4096)

==>   <em>S</em> = 8/3 (1 - 1/4096)

==>   <em>S</em> = 1365/512

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