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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

(a) `alpha-beta`

B

(b) `beta-alpha`

C

(c) `(alpha-beta)/(2)`

D

(d) None of these

Text Solution

Verified by Experts

The correct Answer is:
D
Given that, `sum_(r=1)^(100)a_(2r)=alpha`
` implies a_(2)+a_(4)+"....."+a_(200)=alpha" " "....(i)"`
and `sum_(r=1)^(100)a_(2r-1)=beta`
`implies a_(1)+a_(3)+"......"+a_(199)=beta " " "………(ii)"`
On subtracting Eq.(ii) from Eq.(i), we get
` (a_(2)-a_(1))+(a_(4)-a_(3))+"......."+(a_(200)-a_(199))=alpha -beta`
`d+d+"......"` upto 100 terms `=alpha -beta`
[beacause `a_(n)` be the nth term of AP with common difference d]
`100d=alpha-beta`
`d=(alpha-beta)/(100)`.
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