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 [2026]

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Solution

Q.
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 [2026]

1 −20

2 −26

3 −24

4 −22

Ans.
(4)

$Sum of first 4 terms S_{4} = 6$

$\frac{4}{2} (2 a + 3 d) = 6 \Rightarrow 2 a + 3 d = 3 . . . (1)$

$Sum of first 6 terms S_{6} = 4$

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

$eq. (2) - eq. (1)$

$(2 a + 5 d) - (2 a + 3 d) = \frac{4}{3} - 3$

$\Rightarrow d = - \frac{5}{6}$

$∴$ $2 a + 3 (- \frac{5}{6}) = 3 \Rightarrow a = \frac{11}{4}$

$S_{12} = \frac{12}{2} {2 (\frac{11}{4}) + (12 - 1) (- \frac{5}{6})}$

$S_{12} = 6 (- \frac{22}{6}) = - 22$

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