13. Superfast Techniques for Summation of Numbers

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Some powerful tricks to quickly calculate the sum of numbers from 1 to N,

including cases where numbers increase and then reverse.

I. Sum of First N Natural Numbers

Formula:

S = N(N+1)/2

Ex 1: 1+2+3+...+10

S=10(11)/2=110/2=55

Ans: 55

Ex 2: 1+2+3+...+100

S=100(101)/2=10100/2=5050

 Ans: 5050

Ex 3: 1+2+3+...+1000

S=1000(1001)/2=1001000/2=500500

Ans: 500500

II. Sum of a Series That Reverses (Mirror Addition Technique)

If the series increases and then decreases symmetrically, the sum is simply:

o    Largest Number²

Ex 4: 1+2+1

• Here, the largest number is 2.


• Formula: 2^2 = 4
  Ans: 4

Ex 5: 1+2+3+2+1

• Largest number = 3


• Formula: 3^2 = 9
  Ans: 9

Ex 6: 1+2+3+...+1000+999+...+1

• Largest number = 1000


• Formula: 1000^2 = 1000000
  Ans: 1,000,000
  

  Summary of Tricks

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

  Type of Summation	Formula
  Sum of first N numbers	S = N(N+1)/2
  Increasing & then decreasing series	S=(Largest Number)^2

  
  

  These tricks save time in exams and mental calculations.