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Let a1,a2,.........an be real numbers ...

Let `a_1,a_2,.........a_n` be real numbers such that `sqrt(a_1)+sqrt(a_2-1)+sqrt(a_3-2)++sqrt(a_n-(n-1))=1/2(a_1+a_2+.......+a_n)-(n(n-3)/4` then find the value of `sum_(i=1)^100 a_i`

Text Solution

Verified by Experts

Let `sqrt(a_(i)-(i-1))=b_(i)`
So, we have
`sum_(i=1)^(n)b_(i)=1/2sum_(i=1)^(n)(b_(i)^(2)+(i-1))-(n(n-3))/4`
`rArrsum_(i=1)^(n)b_(i)=1/2sum_(i=1)^(n)b_(i)^(2)+1/2(n(n-1))/2-(n(n-3))/4`
`rArrsum_(i=1)^(n)b_(i)^(2)-2sum_(i=1)^(n)b_(i)+n=0`
`sum_(i=1)^(n)(b_(i)^(2)-2b_(i)+1)=0`
`rArrsum_(i=1)^(n)(b_(i)-1)^(2)=0`
`rArrb_(i)-1=0`
`rArrb_(i)=1`
`rArra_(i)=i`
`thereforesum_(i=1)^(100)=1+2+3+....+100=5050`
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