# 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

The Concepts of number system the mathematics

Divisibility Rules of numbers from 1 to 20 | Basic math education

Simple interest and Compound interest formulas with examples

Percentage formulas | percentage calculations with examples

Circle formulas in math | Area, Circumference, Sector, Chord, Arc of Circle

Types of Triangles With examples | Properties of Triangle

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## 6 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