Sum of Numbers

Sum of First n odd Numbers

Sum of First n odd Numbers  = N²

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Where N = (max no + 1)/2

Q1: What is the sum of first 5 odd numbers?

N = 5;
So the numbers are  1, 3, 5, 7, 9
So N²  = 5² = 25
i.e. 1+3+5+7+9 = 25

Q2: What is the sum of first 6 odd numbers?

N = 6;
So N²  = 6² = 36

Q3: What is the sum of first 60 odd numbers?

N = 60;
So N²  = 60² = 3600

Q4: Find the sum of all odd numbers from 1 to 25?

Now the max number giver here is 25
So N   = (max no + 1)/2 = (25+1)/2 = 13
So N²  = 13² = 169

Sum of First n even Numbers

Sum of First n odd Numbers  = N²  + N
= N(N+1)

Where N = (max no)/2
Q1: What is the sum of first 5 even numbers?

N = 5;
So the numbers are  2, 4, 6, 8, 10
So N² + N = 5² + 5 = 25+5 = 30
i.e. 2+4+6+8+10 = 30

Q2: What is the sum of first 6 even numbers?

N = 6;
So N² +N = 6² +6= 36+6 = 42

Q3: What is the sum of first 45 even numbers?

N = 45;
So N² +N = 45² + 45= 2070

Q4: Find the sum of all odd numbers from 1 to 10?

Now the max number giver here is 25
So N   = (max no)/2 = (10)/2 = 5
So N²  + N = 5² + 5 = 30

Sum of First n Natural Numbers

Sum of First N Natural Numbers  = (N²  + N)/2
= (N * (N+1))/2

Where N is the last Number

Q1: What is the sum of first 5 natural numbers?

N = 5;
So the numbers are  1, 2, 3, 4, 5
So (N² + N)/2 = (5² + 5)/2 = (25+5)/2 = 30/2 = 15
i.e. 1+2+3+4+5 = 15

Q2: What is the sum of first 10 even numbers?

N = 10;
So (N² +N)/2 = (10² + 10)/2 = (100+10)/2 = 110/2 = 55

Q3: What is the sum of Numbers from 1 to 100?

N = 100;
So (N² +N)/2 = (100² + 100)/2 = (10000+100)/2 = 10100/2 = 5050

Sum of First n Cube

Sum of First N Cubes  = ((N²  + N)/2)² 
= ((N * (N+1))/2)² 

Q1: What is the sum of cubes of first 3 natural numbers 3 natural numbers?

N = 3;
So the numbers are  1, 2, 3,  and their cubes are 1, 8, 27 respectively
So ((N² + N)/2)² = ((3² + 3)/2)² = ((9+3)/2)² = (12/2)² = 6² = 36

Q2: What is the sum of first 20 cubes?

N = 20;
So ((N² + N)/2)² = ((20² + 20)/2)² = ((400+20)/2)² = (420/2)² = 210² = 44100

Sum of First n Squares

Sum of First N Squares  = ((N²  + N)*(2N+1))/6

Q1: What is the sum of squares of first 20 natural numbers?

((N²  + N)*(2N+1))/6
=((20²+20)*(2*20+1))/6
=(400+20)*(40+1)/6
=2670

Q2: Find the sum of 1²+2²+3²+.......+30²?

((N²  + N)*(2N+1))/6
=((900²+30)*(2*30+1))/6
=(930*61)/6
=9455

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