Question

the sum of first 6 terms of an ap is 102 amd sum of its first 10 terms is 290. find the ap​

Answer 1

SOLUTION :-

$\boxed{\bf Sum \ of \ n \ terms = \dfrac{n}{2}\times (2a+(n-1)d)}$

$\bullet \sf \ S_6 = 102$

  $\longrightarrow \sf \dfrac{6}{2}\times (2a+(6-1)d)= 102 \\\\\longrightarrow 6\times (2a+5d) = 204 \\\\\longrightarrow 2a+5d =34 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ................(1)$

$\bullet \sf \ S_{10 }= 290$

  $\longrightarrow \sf \dfrac{10}{2}\times (2a+(10-1)d)= 290 \\\\\longrightarrow 10\times (2a+9d) = 580 \\\\\longrightarrow 2a+9d = 58 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ................(2)$

Equation (2) - Equation (1),

$\longrightarrow \sf 4d = 24 \\\\\longrightarrow \bf d = 6$

Substitute the value of d in equation (1),

$\longrightarrow \sf 2a+9\times 6 = 58 \\\\\longrightarrow 2a+54 = 58 \\\\\longrightarrow 2a= 4 \\\\\longrightarrow\bf a = 2$

AP = 2 , 8 , 14 , 20 , 26 , ......................