Question

4. The sum of the first 9 terms of an A.P. is 81 and the sum of its first 20 terms is 400
Find the first term and common difference.
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Answer 1

GIVEN :-

$\bullet \ \sf Sum \ of \ the \ first \ 9 \ terms \ ( S_{9})= 81 \\\\\bullet \ Sum \ of \ the \ first \ 20 \ terms \ (S_{20})= 400$

TO FIND :-

$\bullet$ First term (a) .

$\bullet$ Common difference (d) .

SOLUTION :-

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

$\longrightarrow \sf S_{9} = 81 \\\\\longrightarrow \dfrac{9}{2}\times [ \ 2a+(9-1)d \ ]= 81 \\\\\longrightarrow \dfrac{9}{2}\times (2a+8d) = 81 \\\\\longrightarrow 9 \times (2a+8d) = 162 \\\\\longrightarrow 2a+8d = 18 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ . .........(1)$

$\longrightarrow \sf S_{20}= 400 \\\\\longrightarrow \dfrac{20}{2}\times [ \ 2a+(20-1)d \ ]= 400 \\\\\longrightarrow \dfrac{20}{2}\times (2a+19d)= 400 \\\\\longrightarrow 20\times (2a+19d) = 800 \\\\\longrightarrow 2a+19d = 40 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ . .........(2)$

Equation (2) - Equation (1),

$\longrightarrow \sf 11d = 22 \\\\\longrightarrow \bf d = 2$

Substitute the value of x in equation (1),

$\longrightarrow\sf 2a+ 8 \times 2 = 18 \\\\\longrightarrow 2a + 16 = 18 \\\\\longrightarrow 2a= 2 \\\\\longrightarrow \bf a = 1$

First term = 1

Common difference = 2