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Title : Important Addition Notations Of Natural Number, Strange Reveal In Addition To Fifty-Fifty Reveal Serial
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Important Addition Notations Of Natural Number, Strange Reveal In Addition To Fifty-Fifty Reveal Serial

Dear Students, We are sharing but about summations results of numbers which volition assistance yous inwards solving problems of algebra in addition to trigonometry. Mainly, nosotros focus on amount of release series, amount of foursquare of release serial in addition to amount of cube of release series.

Shortcut / Formula 01: Sum of n Natural Numbers:

1+2+3+4+5+.....................................+n= n(n+1)/2

Example 01: 1+2+3+4+5+6+7+8 =?

Solution: Here n=8

so, 1+2+3+4+5+6+7+8  = 8(8+1)/2 = (8*9)/2 =72/2 = 36

Example 02: 1+2+3+...............+20 =?

Solution: Here n=20

so, 1+2+3+...............+20 = 20(20+1)/2 = (20*21)/2 =420/2 = 240

Shortcut / Formula 02: Sum of n Odd Numbers:

1+3+5+.....................................+n= n(n+1)

Example 01: 1+3+5+7+9+11=?

Solution: Here n=6

so, 1+3+5+7+9+11 = 6(6+1) = (6*7) =42

Example 02: 1+3+5+7+...............+21 =?

Solution: Here n=11

so, 1+3+5+7+...............+21 = 11(11+1) = (11*12) = 132

 

Shortcut / Formula 03: Sum of n Even Numbers:

2+4+6+.....................................+n= n²

Example 01: 2+4+6+8+10+12+14 =?

Solution: Here n=7

so, 2+4+6+8+10+12+14  =  7² = 49

Example 02: 2+4+6+..................+24 = ?

Solution: Here n=12

so, 2+4+6+..................+24  = 12² = 144

Shortcut / Formula 04: Sum of Square of outset n Natural Numbers:

1²+2²+3²+4²+5²+.....................................+n²= [n(n+1)(2n+1)]/6

Example 01: 1²+2²+3²+4²+5²+6²+7²+8² =?

Solution: Here n=8

so, 1²+2²+3²+4²+5²+6²+7²+8²  = [8(8+1)(2*8+1)]/6

= [(8*9)(16+1)]/6 =(72*17)/6 =204

Example 02: 1²+2²+3²+...............+20² =?

Solution: Here n=20

so, 1²+2²+3²+...............+20² = [20(20+1)(20*2+1)]/6

= [(20*21)(40+1)]/6 =(420*41)/6 = 2870

Shortcut / Formula 05: Sum of foursquare of outset n Odd Numbers:

1²+3²+5²+.....................................+n²= n(4n²-1)/3

Example 01: 1²+3²+5²+7²+9²+11²=?

Solution: Here n=6

so, 1²+3²+5²+7²+9²+11²= 6(4*6²-1)/3

= 6(4*36-1)/3 = 2*(144-1)=2*143=286

Example 02: 1²+3²+5²+7²+...............+21² =?

Solution: Here n=11

so, 1²+3²+5²+7²+...............+21²= 11(4*11²-1)/3

= 6(4*121-1)/3 = 2*(484-1)=2*483=966

Shortcut / Formula 06: Sum of foursquare of outset n Even Numbers:

2²+4²+6²+.....................................+n²= [2n(n+1)(2n+1)]/3

Example 01: 2²+4²+6²+8²+10²+12²+14² =?

Solution: Here n=7

so, 2²+4²+6²+8²+10²+12²+14²  =  [2*7(7+1)(2*7+1)]/3

=[14*8(14+1)]/3= [112*15]/3 =112*5 = 560

Example 02: 2²+4²+6²+..................+24² = ?

Solution: Here n=12

so, 2²+4²+6²+..................+24²

=  [2*12(12+1)(2*12+1)]/3

=[24*13(24+1)]/3= [14*13*25]/3 =4550/3 =1516.67

Shortcut / Formula 07: Sum of cube of outset n Natural Numbers:

1³+2³+3³+4³+5³+.....................................+n³= [n(n+1)/2]²

Example 01: 1³+2³+3³+4³+5³+6³+7³+8³ =?

Solution: Here n=8

so, 1³+2³+3³+4³+5³+6³+7³+8³  =[ 8(8+1)/2]²

= [(8*9)/2]² =[72/2]² = 36² =1296

Example 02: 1³+2³+3³+...............+20³ =?

Solution: Here n=20

so, 1³+2³+3³+...............+20³ = [20(20+1)/2]²

= [(20*21)/2]² =[420/2]²= [240]²=57600

Shortcut / Formula 08: Sum of cube of outset n Odd Numbers:

1³+3³+5³+.....................................+n³= 2n²(n+1)²

Example 01: 1³+3³+5³+73+9³+11³=?

Solution: Here n=6

so, 1³+3³+5³+73+9³+11³ = 2*6²(6+1)²

= 2*36(7)² =72*49 =3528

Example 02: 1³+3³+5³+7³+...............+21³ =?

Solution: Here n=11

so, 1³+3³+5³+7³+...............+21³ = 2*11²(11+1)²

=2*121(12)²=242*144 =34848

Shortcut / Formula 09: Sum of cube of outset n Even Numbers:

2³+4³+6³+.....................................+n³= n² (2n²-1)

Example 01: 2³+4³+6³+8³+10³+12³+14³ =?

Solution: Here n=7

so, 2³+4³+6³+8³+10³+12³+14³=  (2*7²-1)7² 

= 49*(2*49-1)=49(98-1)=49*97=4753

Example 02: 2³+4³+6³+..................+24³ = ?

Solution: Here n=12

so, 2³+4³+6³+..................+24³  = (2*12²-1)12² 

= 144(2*144-1)=144(288-1)=144*287=41328

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