IF THE SUM OF FIRST m TERMS OF AN AP IS n AND THE SUM OF FIRST n TERMS - Maths - Arithmetic Progressions

Question

IF THE SUM OF FIRST m TERMS OF AN AP IS n AND THE SUM OF FIRST n
TERMS IS m, THEN SHOW THAT THE SUM OF ITS FIRST( m+n) TERMS IS
-(m+n)

Ankush Jain · Accepted Answer

Given, sum of m terms = n $\Rightarrow s_{m} = n \Rightarrow \frac{m}{n} [2 a + (m - 1) d] = n \Rightarrow 2 a m + m (m - 1) d = 2 n (1)$ Sum of n terms = m $\Rightarrow s_{n} = m \Rightarrow \frac{n}{2} [2 a + (n - 1) d] = m \Rightarrow 2 a n + n (n - 1) d = 2 m (2)$ Subtracting equation (2) from equation (1), we get $2 a m + m (m - 1) d - [2 a n + n (n - 1) d] = 2 n - 2 m \Rightarrow 2 a (m - n) + d [m (m - 1) - n (n - 1)] = 2 (n - m) \Rightarrow 2 a (m - n) + d [m^{2} - m - n^{2} + n] = 2 (n - m) \Rightarrow 2 a (m - n) + d [m^{2} - n^{2} - m + n] = - 2 (m - n) \Rightarrow 2 a (m - n) + d [(m + n) (m - n) - (m - n)] = - 2 (m - n) \Rightarrow 2 a + d (m + n - 1) = - 2 [Taking (m - n) common] 2 a + (m + n - 1) d = - 2 (3)$ $Now, s_{m + n} = (\frac{m + n}{2}) [2 a + (m + n - 1) d] = (\frac{m + n}{2}) (- 2) [using (3)] s_{m + n} = - (m + n)$

IF THE SUM OF FIRST m TERMS OF AN AP IS n. AND THE SUM OF FIRST n TERMS IS m, THEN SHOW THAT THE SUM OF ITS FIRST( m+n) TERMS IS -(m+n).