JEE Main 2026 Jan 22 Shift 1, Mathematics Q20: Arithmetic, Geometric And Harmonic Progressions

JEE Main2026Jan 22, Shift 1Mathematics

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is

Solution

Use $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$ .

$S_{4} = 2 (2 a + 3 d) = 6 \Rightarrow 2 a + 3 d = 3$ …(i)

$S_{6} = 3 (2 a + 5 d) = 4 \Rightarrow 2 a + 5 d = \frac{4}{3}$ …(ii)

(ii) − (i): $2 d = - \frac{5}{3} \Rightarrow d = - \frac{5}{6}$ . Then $2 a = 3 + \frac{5}{2} = \frac{11}{2} \Rightarrow a = \frac{11}{4}$ .

$S_{12} = 6 (2 a + 11 d) = 6 (\frac{11}{2} - \frac{55}{6}) = 6 (\frac{33 - 55}{6}) = - 22$ .

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