Let an be the n^(t h) term of an A.P. If...

Let `a_n` be the `n^(t h)` term of an A.P. If `sum_(r=1)^(100)a_(2r)=alpha&sum_(r=1)^(100)a_(2r-1)=beta,` then the common difference of the A.P. is

` (alpha - beta)/200`

`alpha - beta`

`(alpha-beta)/100`

`beta-alpha`

Text Solution

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The correct Answer is:

Given , `a_(2)+a_(4)+a_(6) +…+a_(200)= alpha " "` …(i)
and `a_(1) +a_(3)+a_(5)+…+a_(199)= beta " "` …(ii)
` :. (a_(2)-a_(1)) +(a_(4)-a_(3))+…+ (a_(200-1_(199)=alpha- beta`
`rArr d + d + d + …..+ 100 "times" = (alpha -beta)`
`rArr d = (alpha - beta )/100`

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Let t_(n) be the nth term of an A.P. If sum_(r = 1)^(10^(99)) a_(2r) = 10^(100) and sum_(r = 1)^(10^(99)) a_(2r - 1) = 10^(99) , then the common difference of A.P. is

`10^(99)`

Let a_(n) be the nth term of a GP of positive integers. If sum_(n=1)^(100) a_(2n) = alpha and sum_(n=1)^(100) a_(2n+1) = beta such that alpha ne beta , then the common ratio is

`(alpha)/(beta)`

`(beta)/(alpha)`

`((alpha)/(beta))^(1//2)`

`((beta)/(alpha))^(1//2)`

Let a_n be nth term of the GP of positive numbers. Let sum_(n=1)^100a_(2n)=alpha and sum_(n=1)^100a_(2n)-1=beta , such that alpha ne beta , then the common ratio is

`alpha/beta`

`beta/alpha`

`sqrt(alpha/beta)`

`sqrt(beta/alpha)`