If the first term of an A.P.

Question

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

Accepted Answer

The first term, $ a = 3 $

Common difference, $ d $

The formula for the sum of the first $ n $ terms of an A.P. is:

$ S_n = \frac{n}{2} [2a + (n-1)d] $

Given:

$ S_4 = \frac{1}{5}(S_8 - S_4) $

This implies:

$ 5S_4 = S_8 - S_4 \quad \Rightarrow \quad 6S_4 = S_8 $

Substituting the sum formulas:

$ 6 \cdot \frac{4}{2}[2 	imes 3 + (4-1)d] = \frac{8}{2}[2 	imes 3 + (8-1)d] $

Simplifying:

$ 6 	imes 2 [6 + 3d] = 4 [6 + 7d] $

$ 12(6 + 3d) = 4(6 + 7d) $

$ 72 + 36d = 24 + 28d $

$ 36d - 28d = 24 - 72 $

$ 8d = -48 $

$ d = -6 $

Now, to find $ S_{20} $:

$ S_{20} = \frac{20}{2} [2 	imes 3 + (20-1)(-6)] $

$ S_{20} = 10 [6 + 19 	imes (-6)] $

$ S_{20} = 10 [6 - 114] $

$ S_{20} = 10 	imes (-108) $

$ S_{20} = -1080 $

Thus, the sum of the first 20 terms is $-1080$.

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