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Question

# The first and last terms of an AP are a and ℓ respectively. If s be the sum of all the terms of the AP, then common difference is

A
2a22S(+a)
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B
2a22S(a)
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C
2+a22S+(+a)
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D
2+a22S(+a)
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Solution

## The correct option is B ℓ2−a22S−(ℓ+a)Let n be the number of terms and snbe the sum of n terms in given AP.Since, l=a+(n−1)d ⇒l−a=(n−1)d ⇒l−an−1=d .....(i)we know sn=n2[a+l] ∴s=n2[a+l] ⇒2s=n(a+l) ⇒n=2sa+lNow put n=2sa+l in equation (i) ⇒l−a2sa+l−1=d⇒l−a2s−(a+l)a+l=d⇒(l−a)(l+a)2s−(a+l)=d⇒d=l2−a22s−(l+a)Option A is correct.

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