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Progression

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New post 05 Oct 2018, 04:10
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If 8 term of an AP is the geometric mean of 1st and 22nd term of the same AP .find the common difference of the AP given that the sum of the first 22 term of the AP is 770?

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Re: Progression  [#permalink]

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New post 05 Oct 2018, 05:32
PBORA1011 wrote:
If 8 term of an AP is the geometric mean of 1st and 22nd term of the same AP .find the common difference of the AP given that the sum of the first 22 term of the AP is 770?

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Let \(a\) be the first term of A.P

\(d=\) Common Difference

n th term \(t_{n}= a+(n-1)d\)

Sum of n term \(=\frac{n[2a+(n-1)d]}{2}\)

According to the question

Sum of \(22\) term \(=\frac{22[2a+(22-1)d]}{2}=770\)

\(2a+21d=70\).......(1)

8 th term \(t_{8}= a+(8-1)d= a+7d\)
22 nd term \(t_{22}= a+(22-1)d= a+21d\)

According to the question
\(a+7d =\sqrt{a(a+21d)}\)

Squaring both sides

\(a^2+49d^2+14ad =a^2+21ad\)
\(49d^2-7ad=0\)
\(7d(7d-a)=0\)
As \(d\neq{0}\), \(7d-a= 0\)........(2)

Combining (1) and (2), we get
\(2(7d)+21d=70\)
\(35d=70\)
\(d=2\)
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Re: Progression   [#permalink] 05 Oct 2018, 05:32
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