If-S-n-is-the-sum-of-the-first-n-terms-of-an-A-P-Express-S-2k-in-terms-of-S-k-and-S-3k-

Question Number 101779 by I want to learn more last updated on 04/Jul/20

If S_{n} is the sum of the first n terms of an A . P .

Express S_{2 k} in terms of S_{k} and S_{3 k}

Answered by bemath last updated on 04/Jul/20

S_{n} = \frac{n (a + u_{n})}{2} \to S_{k} = \frac{k (a + u_{k})}{2}

u_{k} = \frac{2 S_{k}}{k} - a \Rightarrow a + (k - 1) d = \frac{2 S_{k}}{k} - a

(k - 1) d = \frac{2 S_{k}}{k} - 2 a \Rightarrow d = \frac{2 S_{k} - 2 a k}{k (k - 1)} (∙)

S_{2 k} = \frac{2 k (a + u_{2 k})}{2} = k (a + u_{2 k})

u_{2 k} = \frac{S_{2 k}}{k} - a \Leftrightarrow a + (2 k - 1) d = \frac{S_{2 k}}{k} - a

d = \frac{S_{2 k} - a k}{k (2 k - 1)} (∙ ∙)

(∙) = (∙ ∙)

\frac{{2 S}_{k} - 2 a k}{k (k - 1)} = \frac{S_{2 k} - a k}{k (2 k - 1)}

\Rightarrow \frac{(2 k - 1) (2 S_{k} - 2 a k)}{k - 1} + a k = S_{2 k}

Commented by I want to learn more last updated on 05/Jul/20

Thanks sir . I appreciate