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Question

# If ar>0;∀r,n∈N and a1,a2,a3,.....a2n are in A.P, thena1+a2n√a1+√a2+a2+a2n−1√a2+√a3+a3+a2n−2√a3+√a4+...+an+an+1√an+√an+1=

A
n1
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B
n(a1+a2n)a1+an+1
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C
n1a1+an+1
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D
n+1
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Solution

## The correct option is A n(a1+a2n)√a1+√an+1a1,a2,a3,.....a2n are in A.P.Let d be the common difference, then S=2n2(a1+a2n)⇒S=n(a1+a2n)Consider ,a1+a2n√a1+√a2+a2+a2n−1√a2+√a3+a3+a2n−2√a3+√a4+...+an+an+1√an+√an+1=a1+a2n√a1+√a2√a1−√a2√a1−√a2+a2+a2n−1√a2+√a3√a2−√a3√a2−√a3+...+an+an+1√an+√an+1√an−√an+1√an−√an+1=S(√a1−√a2)−nd+S(√a2−√a3)−nd+...+S(√an−√an+1)−nd=S−nd(√a1−√a2+√a2−√a3+.......−√an+√an−√an+1)=S−nd(√a1−√an+1)=(a1+a2n)d(√an+1−√a1)=(a1+a2n)d(√an+1−√a1)(an+1−a1)=(a1+a2n)d(√an+1+√a1)nd=n(a1+a2n)(√an+1+√a1)

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