SUM TO n TERMS OF SPECIAL SERIES

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

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

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

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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