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If $S n$ denotes the sum of first n terms of an AP, then $S 12 = 3 S 8 − S 4$ .

State true (T) or false (F)for the above statement.

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True

False

answer is A.

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Detailed Solution

Given, the sum of first n terms of an AP is Sn.
The sum of the first n terms is represented as,

S n = n 2 2 a + n − 1 d

.
The sum of the first 12 terms,

S 12 = 122 2 a + 12 − 1 d = 6 2 a + 11 d

The sum of the first 8 terms,

S 8 = 82 2 a + 8 − 1 d = 4 2 a + 7 d

The sum of the first 4 terms,

S 4 = 42 2 a + 4 − 1 d = 2 2 a + 3 d

To prove this, lets substitute the values derived in

S 12 = 3 S 8 − S 4

6 2 a + 11 d = 3 4 2 a + 7 d − 2 2 a + 3 d = 3 8 a + 28 d − 4 a − 6 d = 3 4 a + 22 d

6 2 a + 11 d = 3 2 2 a + 11 d = 6 2 a + 11 d

As, LHS=RHS.
Therefore, the given statement, “If

denotes the sum of first n terms of an AP, then

.” is proved to be true.

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