Find the common difference of an A.P. whose f | vedclass

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(B) Let the first term be $a = 100$ and the common difference be $d$.The sum of the first $n$ terms of an $A.P.$ is given by $S_n = \frac{n}{2}[2a + (

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$-10$

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(B) Let the first term be $a = 100$ and the common difference be $d$.The sum of the first $n$ terms of an $A.P.$ is given by $S_n = \frac{n}{2}[2a + (n-1)d]$.The sum of the first six terms is $S_6 = \frac{6}{2}[2(100) + 5d] = 3(200 + 5d) = 600 + 15d$.The sum of the next six terms is the sum of the first $12$ terms minus the sum of the first $6$ terms: $S_{12} - S_6$.$S_{12} = \frac{12}{2}[2(100) + 11d] = 6(200 + 11d) = 1200 + 66d$.Sum of the next six terms $= (1200 + 66d) - (600 + 15d) = 600 + 51d$.According to the problem,$S_6 = 5 	imes (	ext{sum of next six terms})$.$600 + 15d = 5(600 + 51d)$.$600 + 15d = 3000 + 255d$.$15d - 255d = 3000 - 600$.$-240d = 2400$.$d = \frac{2400}{-240} = -10$.

Find the common difference of an $A.P.$ whose first term is $100$ and the sum of whose first six terms is five times the sum of the next six terms.

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