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If in an A.P. the sum of m terms is equal to n and the sum of

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If in an A.P. the sum of $m$ terms is equal to $n$ and the sum of $n$ terms is equal to $m$ , then prove that the sum of $(m + n)$ terms is $- (m + n)$ .

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Mateo Discussed

If in an A.P. the sum of

m

terms is equal to

n

and the sum of

n

terms is equal to

m

, then prove that the sum of

(m + n)

terms is

- (m + n)

5 mins ago

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Text solutionVerified

Let

a

be the first term and

d

be the common difference of the given A.P. Then,

S_{m} = n

⟹ \frac{m}{2} {2 a + (m - 1) d} = n

⟹ 2 am + m (m - 1) d = 2 n

...(i)

and,

S_{n} = m

⟹ \frac{n}{2} {2 a + (n - 1) d}

⟹ 2 an + n (n - 1) d = 2 m

...(ii)

Subtracting equation (ii) from equation (i), we get

2 a (m - n) + {m (m - 1) - n (n - 1)} d = 2 n - 2 m

⟹ 2 a (m - n) + {(m^{2} - n^{2}) - (m - n)} d = - 2 (m - n)

⟹ 2 a + (m + n - 1) d = - 2

[On dividing both sides by

(m - n)

] ...(iii)

Now,

S_{m + n} = \frac{m + n}{2} {2 a + (m + n - 1) d}

⟹ S_{m + n} = \frac{m + n}{2} (- 2)

[Using (iii)]

⟹ S_{m + n} = - (m + n)

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Question Text	If in an A.P. the sum of $m$ terms is equal to $n$ and the sum of $n$ terms is equal to $m$ , then prove that the sum of $(m + n)$ terms is $- (m + n)$ .
Answer Type	Text solution:1
Upvotes	150