radicals are fun....
Algebra
Level
2
Find the value of n such that 1/[sqrt(1)+sqrt(2)] + 1/[sqrt(2)+sqrt(3)] + 1/[sqrt(3)+sqrt(4)] + ... + 1/[sqrt(n)+sqrt(n+1)] = 10.
The answer is 120.
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1 solution
1/(sq.rt(1) + sq.rt(2)) + 1/(sq.rt(2) + sq.rt(3)) + 1/(sq.rt(3) + sq.rt(4)) + ... + 1/(sq.rt(n) + sq.rt(n+1)) = 10
Rationalize the denominator...
(sq.rt(1) - sq.rt(2))/(-1) + (sq.rt(2) - sq.rt(3))/(-1) + (sq.rt(3) - sq.rt(4))/(-1) + ... + (sq.rt(n) - sq.rt(n+1))/(-1) = 10
So, all are being cancel... Except for sq.rt(1) and -(sq.rt(n+1))
(sq.rt(1) - sq.rt(n+1))/(-1) = 10
(1 - sq.rt(n+1)) = -10
-1 + sq.rt(n+1) = 10
sq.rt(n+1) = 11
n+1 = 121
n = 120...