If the sum of the first four terms of an A.P.

Question

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

Accepted Answer

Let the A.P. have first term $a$ and common difference $d$.

$$S_4 = 4a+6d = 6 \implies 2a+3d=3, \qquad S_6 = 6a+15d = 4.$$

Solving these two equations: $d = -\dfrac{5}{6}$, $a = \dfrac{11}{4}$.

$$S_{12} = \frac{12}{2}\left[2a+11d\right] = 6\left[\frac{11}{2} - \frac{55}{6}\right] = -22.$$

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