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Let a(n) be the nth term of an AP, if su...

Let `a_(n)` be the nth term of an AP, if `sum_(r=1)^(100)a_(2r)=alpha " and "sum_(r=1)^(100)a_(2r-1)=beta`, then the common difference of the AP is

A

`(alpha-beta)/(200)`

B

`alpha-beta`

C

`(alpha-beta)/(100)`

D

`beta-alpha`

Text Solution

Verified by Experts

The correct Answer is:
C
Given, `a_(2)+a_(4)+a_(6)+"......."+a_(200)=alpha" " "....(i)"`
and `a_(1)+a_(3)+a_(5)+"......."+a_(199)=beta" " "....(ii)"`
On subtracting Eq. (ii) from Eq. (i), we get
`(a_(2)-a_(1))+(a_(4)-a_(3))+(a_(6)-a_(5))+"......."+(a_(200)-a_(199))=alpha-beta`
`implies d+d+d+"....."+d=alpha-beta implies 100d=alpha-beta`
`:. d=((alpha-beta))/(100)`.
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