Let `lt a_n gt `be an arithmetic sequence of 99 terms such that sum of its odd numbered terms is 1000 then the value of
`Sigma_(r=1)^(50) (-1)^((r(r+1))/2).a_(2r-1)` is _________.

Answer

(-40)

Explanation

Given that `a_(1)+a_(3)+..+a_(99)=1000`
`rArr25(a_(1)+a_(99))=1000`
`rArra_(1)+a_(99)=40`
Now, `S=sum_(r=1)^(50)(-1)^((r(r+1))/2)cdota_(2r-1)`
`=-a_(1)-a_(3)+a_(5)+a_(7)-a_(9)-a_(11)+..+a_(93)+a_(95)-a_(97)-a_(99)`
There are 26 negative terms and 24 positive terms.
`thereforeS=-a_(1)-a_(99)=-40`