# Sum of n Consecutive numbers Like Natural, Even, Odd, Squares, Cubes

## Formulas with examples for Sum of n Consecutive numbers

In this page provide formulas with examples for sum of n consecutive natural numbers, sum of positive integers,  sum of n odd and even numbers, sum of consecutive squares of natural, odd, even numbers, sum of consecutive cube of natural, odd, even numbers.

### Sum of natural, odd & even numbers

Sum of “n” natural numbers = Sum of “n” natural even numbers = (n ) (n + 1)

Sum of “n” natural odd numbers = n 2

### Sum of square of natural, odd & even numbers

Sum of square of “n” first or consecutive square natural numbers: Sum of square of “n” first or consecutive odd numbers: Sum of square of “n” first or consecutive square even numbers: ### Sum of cube natural, odd & even numbers

Sum of cube of first or consecutive ” n” natural numbers: Sum of cube of first or consecutive ” n” even natural numbers = 2n2 (n + 1)2
Sum of cube of first or consecutive ” n” odd natural numbers = n2 (2n2 – 1)

### Examples on sum of numbers

Ex . 1 : Find the sum of the first 50 positive integers.

Sol:   1 + 2 + 3+ 4+ 5+ ———-+50 So Here n = 50

= 50 ( 50+1) / 2 = 25 x 51 = 1275

Ex . 2 : Find the sum of the  consecutive numbers 25+26+27+28+ —–+100 .

Sol: 25+26+27+28+ —–+50 = ( 1+2+3+4+———+100) – (1+2+3+4+——-24)

= [ 100 ( 100+1) / 2 ] – [ 24 ( 24+1) / 2 ]

= 5050 – 300 = 4750.

Ex . 3 : Find the sum of the squares of the first 60 natural numbers.

Sol: 12 + 22 + 32 + 42+ 52 + ———-+602  So Here n = 60

= { 60 x (60 + 1) x [( 2 x 60 )+1 ] } / 6

= 60 x 61 x 121 / 6

=73810

Ex . 4 : what is the sum of first 100 odd numbers?

Sol : first 100 odd numbers means  1 + 3 + 5 +7 + ———-+ 199 so here n = 100

= 1002 = 10000

Ex . 5 : Find the sum of consecutive odd numbers  51 +53 +55 + ———+ 199.

Sol : 51 +53 +55 + ———+ 199 = {1 + 2+ 3  ———+ 199} – { 1 +2 + 3 + ———+ 49}

=1002  – 25

= 10000- 625 = 9375 .

Ex . 6 : Find the sum of the cubes of the first 25 positive integers.

Sol: 13 + 23+ 33+ 43+ 53 + ———-+253 So Here n = 25

= 252 x (25 +1 )2 / 4

= 625 x 676 / 4 = 105625

Ex . 6 : Find the sum of the cubes of the first 25 odd numbers.

Sol: First 25 odd cube numbers means 13 + 33+ 53 + ———-+493 So Here n = 25

= 25 2 [ (2 x  252 )– 1 ]

= 625 x [ 1250 – 1]

=625 x 1249 = 780625

Ex . 7 : Find the sum of the  consecutive cube numbers 263+283+ 303 + 323—–+1003 .

Sol : 263+283+ 303 + 323—–+1003 = {23+43+ 63 + 83—–+1003 }  – {23+43+ 63 + 83—–+243}

=(2 x 502 )(50 + 1)–  (2 x 122 )(12 + 1)2

= [ 5000 x 2601] –  [ 288 x 169 ]

= 13005000 – 48672 = 12956328.

Ex . 8 : Find the sum of the  consecutive square odd numbers 752 + 772 + 792 + 812+ 832 + ———-+992 .

Sol : 752 + 772 + 792 + 812+ 832 + ———-+992  = {12 + 22 + 32+—–+992 } – { 12 + 22 + 32 + —–+732}

= [ ( 50 ) (4*50 2 – 1) / 3 ] – [ ( 37 ) (4*37 2 – 1) / 3 ]

= [ 50  x 9999 / 3 ] – [ 37 x 5475 / 3]

= 166650 – 67525 = 99125 Some related Topics in Quantitative aptitude

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## 7 thoughts on “Sum of n Consecutive numbers Like Natural, Even, Odd, Squares, Cubes” #### Sai Venkatesh malakala

(July 8, 2018 - 8:20 am)

Thanks a lot #### sivaalluri

(July 9, 2018 - 2:09 pm)

Thank you Mr.Sai Venkatesh malakala #### Bharathi

(April 9, 2019 - 2:06 am)

Thank you sir, please provide more short cut tricks for competitive exams #### sivaalluri

(April 20, 2019 - 3:15 pm)

Definitely we will upload one by one #### Ravindra Nath Mahto

(May 11, 2019 - 5:33 am)

Thank you so much sir… #### sivaalluri

(May 11, 2019 - 3:43 pm)

Thank you #### Vijay

(April 16, 2023 - 8:08 am)

Thanks for the short trick, need your help to solve below question.
Sum of cubes of six consecutive numbers is 15471. Find the sum of cubes of odd numbers and square of even numbers.