1 + 2 + 3 + ............ + n = n(n + 1)/2
Example 1 :
Find the sum :
1 + 2 + 3 + ............ + 60
Solution :
Sum of first n natural numbers :
1 + 2 + 3 + ............ + n = n(n + 1)/2
Substitute n = 60.
1 + 2 + 3 + ............ + 60 = 60(60 + 1)/2
= 30(61)
= 1830
Example 2 :
Find the sum :
3 + 6 + 9 + ............ + 96
Solution :
= 3 + 6 + 9 + ............ + 96
= 3(1 + 2 + 3 + ............ + 32)
= 3[32(32 + 1)/2]
= 3(16)(33)
= 1584
Example 3 :
Find the sum :
51 + 52 + 53 + ............ + 92
Solution :
51+ 52 + 53 + ............ + 92 :
= (1 + 2 + 3 + ............ + 92) - (1 + 2 + 3 + ............ + 50)
= [92(92 + 1)/2] - [50(50 + 1)/2]
= 46(93) - 25(51)
= 4278 - 1275
= 3003
12 + 22 + 32 + ............ + n2 = [n(n + 1)(2n + 1)]/6
Example 4 :
Find the sum :
1 + 4 + 9 + 16 + ............ + 225
Solution :
1 + 4 + 9 + 16 + ............ + 225 :
= 12 + 22 + 32 + 42 + ............ + 152
Sum of squares of first n natural numbers :
12 + 22 + 32 + 42 + ............ + n2 = [n(n + 1)(2n + 1)]/6
Substitute n = 15
12 + 22 + 32 + 42 + ............ + 152 :
= [15(15 + 1)(2(15) + 1)]/6
= 15(16)(31)/6
= 5(8)(31)
= 1240
Example 5 :
Find the sum :
62 + 72 + 82 + ............ +212
Solution :
62 + 72 + 82 + ............ + 212 :
= (12 + 22 + ............ + 212) - (12 + 22 + ............ + 52)
= [21(21 + 1)(2(21) + 1)/6] - [5(5 + 1)(2(5) + 1)/6]
= [21(22)(43)/6] - [5(6)(11)/6]
= [7(11)(43)] - [5(1)(11)]
= 3311 - 55
= 3256
13 + 23 + 33 + ............ + n3 = [n(n + 1)/2]2
Example 6 :
103 + 113 + 123 + ............ + 203
Solution :
103 + 113 + 123 + ............ + 203 :
= (13 + 23 + 33 +.........+ 203) - (13 + 23 + 33 +.........+ 93)
Sum of cubes of first n natural numbers :
13 + 23 + 33 +.........+ n3 = [n(n + 1)/2]2
Then,
= [20(20 + 1)/2]2 - [9(9 + 1)/2]2
= 2102 - 452
= (210 + 45)(210 - 45)
= 255(165)
= 42075
When number of terms n is given :
1 + 3 + 5 + ............ to n terms = n2
When the last term l is given :
1 + 3 + 5 + ............ + l = [(l + 1)/2]2
Example 7 :
1 + 3 + 5 + ............ to 25 terms
Solution :
Sum of first n odd numbers :
1 + 3 + 5 + ............ + n = n2
Substitute n = 25.
= 252
= 625
Example 8 :
1 + 3 + 5 + ............ + 71
Solution :
Sum of first n odd numbers :
1 + 3 + 5 + ............ + l = [(l + 1)/2]2
Substitute l = 71.
= [(71 + 1)/2]2
= [72/2]2
= 362
= 1296
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