JEE Main 2025 Jan 23 Shift 1, Mathematics Q8: Arithmetic, Geometric And Harmonic Progressions

JEE Main2025Jan 23, Shift 1Mathematics

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

A
$- 1200$
B
$- 1080$
C
$- 1020$
D
$- 120$

Solution

Let the common difference be $d$ and $a = 3$ .

The 'sum of the next four terms' is $S_{8} - S_{4}$ . The given condition is $S_{4} = \frac{1}{5} (S_{8} - S_{4})$ , i.e. $6 S_{4} = S_{8}$ .

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

$6 \cdot 2 [6 + 3 d] = 4 [6 + 7 d] \Rightarrow 12 (6 + 3 d) = 4 (6 + 7 d)$

$\Rightarrow 18 + 9 d = 6 + 7 d \Rightarrow d = - 6$ .

$S_{20} = \frac{20}{2} [2 \cdot 3 + 19 (- 6)] = 10 [6 - 114] = - 1080$ .

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JEE Main 2025 Jan 23 Shift 1, Mathematics Q8: Arithmetic, Geometric And Harmonic Progressions

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