## Formulas with examples for Sum of n Consecutive 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 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**=

*2n*

**(n + 1)**^{2}^{2}**Sum of cube of first or consecutive ” n” odd natural numbers**=

*n*

**(2n**^{2}**– 1)**^{2}### 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: 1* ^{2}* + 2

*+ 3*

^{2}*+ 4*

^{2}*+ 5*

^{2}*+ ———-+60*

^{2}*So Here n = 60*

^{2}= { 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

= 100* ^{2}* = 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}

=100* ^{2}* – 25

^{2 }= 10000- 625 = 9375 .

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

Sol: 1* ^{3}* + 2

*+ 3*

^{3}*+ 4*

^{3}*+ 5*

^{3}*+ ———-+25*

^{3}*So Here n = 25*

^{3}= 25* ^{2}* x (25 +1 )

*/ 4*

^{2}= 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 1* ^{3}* + 3

*+ 5*

^{3}*+ ———-+49*

^{3}*So Here n = 25*

^{3}= *25 ^{2} [ (2 x 25^{2} )– 1 ]*

= 625 x [ 1250 – 1]

=625 x 1249 = 780625

Ex . 7 : Find the sum of the consecutive cube numbers 26* ^{3}*+28

*+ 30*

^{3}*+ 32*

^{3}*—–+100*

^{3}*.*

^{3}Sol : 26* ^{3}*+28

*+ 30*

^{3}*+ 32*

^{3}*—–+100*

^{3}*= {2*

^{3}*+4*

^{3}*+ 6*

^{3}*+ 8*

^{3}*—–+100*

^{3}

^{3}} – {2^{3}+4^{3}+ 6^{3}+ 8^{3}—–+24^{3}}=(2 x 50** ^{2}** )(50 + 1)

**– (2 x 12**

^{2 }**)(12 + 1)**

^{2}

^{2 }= [ 5000 x 2601] – [ 288 x 169 ]

= 13005000 – 48672 = 12956328.

Ex . 8 : Find the sum of the consecutive square odd numbers 75* ^{2} + 77^{2} + 79^{2} + 81^{2}+ 83^{2} + ———-+99^{2}* .

Sol : 75* ^{2} + 77^{2} + 79^{2} + 81^{2}+ 83^{2} + ———-+99^{2}* = {1

*} – { 1*

^{2}+ 2^{2}+ 3^{2}+—–+99^{2}*}*

^{2}+ 2^{2}+ 3^{2}+ —–+73^{2}= [ ( 50 ) (4*50 * ^{2}* – 1) / 3 ] – [ ( 37 ) (4*37

*– 1) / 3 ]*

^{2}= [ 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 Quadrilateral | Quadrilateral formula for area and perimeter

Types of Triangles With examples | Properties of Triangle

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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.