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

Options

A. `alpha-beta`
B. `beta-alpha`
C. `(alpha-beta)/(2)`
D. None of these

Answer

D

Explanation

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)`.