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In an A.P., of which a is the first term, if the sum of the fi

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In an A.P., of which a is the first term, if the sum of the first p terms is zero, show that the sum of the next q terms is $- \frac{a ( p + q ) q}{p - 1}$ .

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In an A.P., of which a is the first term, if the sum of the first p terms is zero, show that the sum of the next q terms is

- \frac{a ( p + q ) q}{p - 1}

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Text solutionVerified

S_{p} = 0

∴ \frac{P}{2} [2 a + (p - 1) d] = 0

∴ d = \frac{- 2 a}{p - 1}

(1)

Sum of next q terms

=

sum of an A.P. whose first term will be

T_{p + 1} = a + p d

∴ S = \frac{q}{2} [2 (a + p d) + (q - 1) d]

= \frac{q}{2} [2 a + (p - 1) d + (p + q) d]

= \frac{q}{2} [0 - (p + q) \frac{2 a}{p - 1}] = - a \frac{( p + q ) q}{p - 1}

, by

(1)

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Question Text	In an A.P., of which a is the first term, if the sum of the first p terms is zero, show that the sum of the next q terms is $- \frac{a ( p + q ) q}{p - 1}$ .
Answer Type	Text solution:1
Upvotes	150