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Question

The sum of first n terms of an A.P. whose first terms is 8 and the common difference is 20 is equal to the sum of first 2n terms of another A.P. whose first term is -30, and the common difference is 8. Find n. 


Solution

For $$1^{st}$$  AP
$$a = 8, d= 20$$
For $$2^{nd}$$ AP
$$a' = -30 , d' = 8$$
According to the question, $$S_n  = S_{2n}'$$
$$\Rightarrow \dfrac{n}{2}[2a+(n-1)d] = \dfrac{2n}{2}[2a' + (2n-1)d']$$
$$\Rightarrow [2(8) + (n-1)20]  = 2[2(-30)+(2n-1)8]$$
$$\Rightarrow 2 \times 8 + n \times 20 - 20 = 2 [-60 + 16n -8]$$
$$\Rightarrow 20 n - 4  = -136 + 32 n$$
$$\Rightarrow -32 n+ 20 n  = -136 + 4$$
$$\Rightarrow n = 11$$
Hence, the required value of n is 11.

Maths

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