+208 If {\(a_n\)} is an arithmetic sequence, \(a_1\)= 1, and \(a_{100}\)= 625, evaluate:
\(1\over{\sqrt{a_1}+\sqrt{a_2}}\)+\(1\over{\sqrt{a_2}+\sqrt{a_3}}\)+\(1\over{\sqrt{a_3}+\sqrt{a_4}}\)+...+\(1\over{\sqrt{a_{99}}+\sqrt{a_{100}}}\)
+128570 Note
1/ [sqrt a2 + sqrt a1 ] if we multiply top/bottom by the conjugate we have
[ sqrt a2 - sqrt a1 ] / [ a2 -a1 ] = [ sqrt a2 - sqrt 1 ] / d
Also
1/ [sqrt a3 + sqrt a2 ] if we multiply top/bottom by the conjugate we have
[ sqrt a3 - sqrt a2 ] / [ a3 -a2 ] = [ sqrt a3 - sqrt a2 ] / d
So we have [sqrt a2 - sqrt a1 ] / d + [sqrt a3 - sqrt a2 ] /d + [ sqrt a4 -sqrt a3]/ d +.....+
[sqrt a99 - sqrt a98 ] /d + [sqrt a100 - sqrt a99] / d
So....eventually we just end up with
[ -sqrt a1 + sqrt a100 ] / d = [ -1 + 25 ] / d
We can find d by noting that
a1 + 99d = a100
1 + 99d = 625
624 =99d
624/99 = d =208/ 33
So...we have
[ 24 ] / [ 208/33] = 24 * 33 / 208 = 99 / 26
