In an A.P., the sum of the first 10 terms is – 150, and the sum of the next 10 terms is – 550. Find the A.P.

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Solution

From the question it is given that,
The sum of the first 10 terms $= - 150$
The sum of the next 10 terms $= - 550$
A.P $= ?$
We know that, $S_{n} = (n / 2) [2 a + (n - 1) d]$
$\begin{matrix} S_{10} = (n / 2) [2 a + (10 - 1) d] - 150 = (10 / 2) [2 a + 9 d] - 150 = 5 [2 a + 9 d] - 150 = 10 a + 45 d \end{matrix}$
Then, $S_{20} = S_{10} + S_{10}$
$\begin{matrix} \begin{matrix} = & - 150 - 550 = & - 700 \end{matrix} \begin{matrix} S_{20} = (20 / 2) [2 a + 19 d] - 700 = 10 (2 a + 19 d) - 700 = 20 a + 190 d \end{matrix} \end{matrix}$

Now, multiplying equation(i) by 2 we get,
$20 a + 90 d = - 300$
Subtract equation (iii) from equation (ii),
$(20 a + 190 d) - (20 a + 90 d) = - 700 - (- 300)$
$\begin{matrix} 20 a + 190 d - 20 a - 90 d = - 700 + 300 100 d = - 400 d = - 400 / 100 \end{matrix}$
$d = - 4$

Substitute the value of d in equation (i) we get,
$\begin{matrix} 10 a + 45 (- 4) = - 150 10 a - 180 = - 150 10 a = - 150 + 180 10 a = 30 a = 30 / 10 a = 3 a_{2} = 3 + (- 4) = 3 - 4 = - 1 a_{3} = - 1 + (- 4) = - 1 - 4 = - 5 a_{3} = - 5 + (- 4) = - 5 - 4 = - 9 \end{matrix}$
Therefore, A.P. is 3, -1, -5, -9, …

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In an AP ,the sum of 10terms is 150 and sum of next 10 term is 550

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