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avatar+214 

If {\(a_n\)} is an arithmetic sequence, \(a_1\)= 1, and \(a_{100}\)= 625, evaluate:

\(1\over{\sqrt{a_1}+\sqrt{a_2}}\)+\(1\over{\sqrt{a_2}+\sqrt{a_3}}\)+\(1\over{\sqrt{a_3}+\sqrt{a_4}}\)+...+\(1\over{\sqrt{a_{99}}+\sqrt{a_{100}}}\)

 #1
avatar+128794 
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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

 

 

cool cool cool

 Apr 7, 2024
 #2
avatar+214 
+1

Thanks a lot!smiley


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