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If Sn1 denotes the sum of first n terms of an A.P., prove that S12 = 3(S8 − S4). - Mathematics

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If Sn1 denotes the sum of first n terms of an A.P., prove that S12 = 3(S8 − S4).

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Solution

Let a and d be the first term and the common difference of AP, respectively.

We know

`S_n=n/2[2a+(n-1)d]`

`:.S_8=8/2[2a+(8-1)d] " and "S_4= 4/2[2a+(4-1)d]`

S8=4(2a+7d) and S4=2(2a+3d)

S8=8a+28d and S4=4a+6d

Now,

3(S8S4)=3(8a+28d4a6d)

= 3(4a+22d)

= 6(2a+11d)

`=12/2[2a+(12-1)d]`

= S12

∴ S12=3(S8S4)

Concept: Sum of First n Terms of an AP
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