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

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  – 25² 

= 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  –  (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