If a1,a2,a3,....an are in A.P., where a10 for all i, then a1+a21+a2+a31+....+an−1+an1 is equal to

Question

If a1​,a2​,a3​,....an​ are in A.P., where a1​>0 for all i, then a1​​+a2​​1​+a2​​+a3​​1​+....+an−1​​+an​​1​ is equal to

Filo · Accepted Answer

a1​,a2​,a3​........A.P.,Soa2​−a1​=a3​−a2​=a4​−a3​=d​.....(i)Now,given series:-a1​​+a2​​1​+a2​​+a3​​1​+a3​​+a4​​1​+−−−−−−−−−+an−1​​+an​​1​=a1​​+a2​​1​×a2​​−a1​​a2​​−a1​​​+a3​​+a2​​×(a3​​−a2​​)a3​​−a2​​​+−−−−−−−−+an−1​​+an​​×(an−1​​−an​​)an−1​​−an​​​=a2​−a1​a2​​−a1​​​+a3​−a2​a3​​−a2​​​+−−−−−−an​​−an−1​​an​​−an−1​​​=d1​[a2​​−a1​​+a3​​−a2​​+−−−−−−−+an​​−an−1​​]........................from (i)=d1​[a2​​−a1​​+a3​​−a2​​+a4​​−a3​​+−−−−−+an−1​​−an−2​​+an​​−an+1​​]=[an​​−a1​​]×d1​................(ii)now,an​=a1​+(n−1)dan​−a1​=(n−1)d⇒d=n−1an​−a1​​.......(iii)replacing it in (ii)=(an​−a1​)n−1​×(an​​−a1​​)=(an​​+a1​​)n−1​.........(D)

Question Text	If $a_{1}, a_{2}, a_{3}, .... a_{n}$ are in A.P., where $a_{1} > 0$ for all $i$ , then $\frac{1}{a _{1} + a _{2}} + \frac{1}{a _{2} + a _{3}} + .... + \frac{1}{a _{n - 1} + a _{n}}$ is equal to
Answer Type	Text solution:1
Upvotes	150

If $a_{1}, a_{2}, a_{3}, .... a_{n}$ are in A.P., where $a_{1} > 0$ for all $i$ , then $\frac{1}{a _{1} + a _{2}} + \frac{1}{a _{2} + a _{3}} + .... + \frac{1}{a _{n - 1} + a _{n}}$ is equal to

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If a1​,a2​,a3​,....an​ are in A.P., where a1​>0 for all i, then a1​​+a2​​1​+a2​​+a3​​1​+....+an−1​​+an​​1​ is equal to

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If $a_{1}, a_{2}, a_{3}, .... a_{n}$ are in A.P., where $a_{1} > 0$ for all $i$ , then $\frac{1}{a _{1} + a _{2}} + \frac{1}{a _{2} + a _{3}} + .... + \frac{1}{a _{n - 1} + a _{n}}$ is equal to