## Formulas with examples for Sum of n Consecutive numbers

Contents

### 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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## 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