Algebra ─ Telescoping Series : 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 + 1/110 = ?

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Algebra
 └─ Sequence & Series
   └─ Telescoping Series

 

Trick for: 1/3*4 + 1 / 4*5 + 1/5*6 + ⋯ + 1/10*11

Trick Formula for Series: 1/n(n+1) = 1/n - 1/n+1

Final Shortcut Memory Line: "Start with 1/n, end with 1/(n+k). Ans = First - Last"​

Apply Trick:

"Start from the first 'n', end at the last '(n+k)'. Ans = First - Last"

 So directly: =1/3 - 1/11=11 - 3 /33 = 8/33 ​

Simply,

= Last Term - 1st Term / Last Term * 1st Term 

= 11 - 3/11*3

= 8/33