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 [2025]

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Solution

Q.
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 [2025]

1 –120

2 –1200

3 –1020

4 –1080

Ans.
(4)

Let first term be 'a' and common difference be 'd'.

Also, $S_{4} = \frac{1}{5} (S_{8} - S_{4})$

$\Rightarrow 5 S_{4} = S_{8} - S_{4} \Rightarrow S_{8} = 6 S_{4}$

$\Rightarrow \frac{8}{2} [2 \times 3 + 7 \times d] = 6 \times \frac{4}{2} [2 \times 3 + 3 \times d]$ [ $∵ a = 3$ ]

$\Rightarrow 6 + 7 d = 18 + 9 d \Rightarrow 2 d = - 12 \Rightarrow d = - 6$

$∴ S_{20} = \frac{20}{2} [2 \times 3 + (19) \times - 6]$

= 10[– 108] = – 1080.

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