Can You Simplify This Radical Expression ? | Math Olympiad

Can You Simplify This Radical Expression ? | Math Olympiad
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[@RyanLewis-Johnson-wq6xs]

Sqrt[1/5+1/45+1/117+1/221+1/357+1/525]=0.2Sqrt[6]=Sqrt[6]/5

[@harrymatabal8448]

Excellent

[@rcnayak_58]

A Nice way. But we can still solve in a simpler way. We can observe the difference between the two numbers in the denominator of each term, which is 4 (we can write 1/5 in the first term as 1/(1x5)). Now here comes the trick. Let us multiply the whole expression by 4/4. So that it will be (1/4) [ {4 / (1x5)} + {4/(5x9)} + {4 /(9 x13)} + {4 /(13 x 17)} + {4 /(17 x 21)} + {4 /(21 x 25)}]. This 4 appearing in the numerator can be rewritten as (1/4) [ {(5-1) / (1x5)} + {(9-5)/(5x9)} + {(13-9) / (9 x 13)} + {(17-13)/ (13 x 17)} + {(21-17)/ (17 x 21)} + {(25-21)/ (21 x 25)}]. On expanding it, we get (1/4)[ {1 - (1/5)} + {(1/5) - (1/9)} + {(1/9) - (1/13} + { (1/13) - (1/17)} + {(1/17) - (1/21)} + { (1/21) - (1/25)}]. This reduces to only (1/4) [ 1- (1/25)] , which on simplification becomes 6/25. Since there is a square root involved, the result becomes √6/5.