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If a(1),a(2),….a(n) are in arthimatic pr...

If `a_(1),a_(2),….a_(n)` are in arthimatic progression, where `a_(i)gt0` for all I, then show that
`1/(sqrt(a_(1))+sqrt(a_(2)))+1/(sqrt(a_(2))+sqrt(a_(3)))+…+1/(sqrt(a_(n-1))+sqrt(a_(n)))`
`(n-1)/(sqrt(a_(1))+sqrt(a_(n)))`

Text Solution

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Since, `a_(1), a_(2),...,a_(n)` are in an AP
`:. (a_(2) -a_(1)) = (a_(3) - a_(2)) = .. = (a_(n) - a_(n-1)) = d`
Thus, `(1)/(sqrt(a_(1)) + sqrt(a_(2))) + (1)/(sqrt(a_(2)) + sqrt(a_(3))) + ...+ (1)/(sqrt(a_(n-1)) + sqrt(a_(n)))`
`= ((sqrt(a_(2)) - sqrt(a_(1)))/(d)) + ((sqrt(a_(3)) - sqrt(a_(2)))/(d)) + ....+ ((sqrt(a_(n)) - sqrt(a_(n-1)))/(d))`
`= (1)/(d) (sqrt(a_(n)) - sqrt(a_(1))) = (1)/(d) ((a_(n) -a_(1)))/(sqrt(a_(n)) + sqrt(a_(1))) = ((n-1))/(sqrt(a_(n)) + sqrt(a_(1)))`
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