SUM OF SPECIAL SERIES WORKSHEET

Find the sum of the following series :

Problem 1 :

1 + 2 + 3 + ............ + 60

Problem 2 :

3 + 6 + 9 + ............ + 96

Problem 3 :

51 + 52 + 53 + ............ + 92

Problem 4 :

1 + 4 + 9 + 16 + ............ + 225

Problem 5 :

6² + 7² + 8² + ............ +21²

Problem 6 :

10³ + 11³ + 12³ + ............ + 20³

Problem 7 :

1 + 3 + 5 + ............ to 25 terms

Problem 8 :

1 + 3 + 5 + ............ + 71

Problem 9 :

19 + 20 + 21 + ............ + 325

Problem 10 :

2 + 6 + 12 + 20 ............ + 650

Answers

1. Answer :

1 + 2 + 3 + ............ + 60

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

2. Answer :

= 3 + 6 + 9 + ............ + 96

= 3(1 + 2 + 3 + ............ + 32)

= 3[32(32 + 1)/2]

= 3(16)(33)

= 1584

3. Answer :

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

4. Answer :

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

5. Answer :

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

6. Answer :

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

7. Answer :

1 + 3 + 5 + ............ to 25 terms

Formula :

1 + 3 + 5 + ............ + to n terms = n²

Substitute n = 25.

= 25²

= 625

8. Answer :

1 + 3 + 5 + ............ + 71

Formula :

1 + 3 + 5 + ............ + l = [(l + 1)/2]²

Substitute l = 71.

= [(71 + 1)/2]²

= [72/2]²

= 36²

= 1296

9. Answer :

19 + 20 + 21 + ............ + 325 :

= (1 + 3 + 5 + ............ + 325) - (1 + 3 + 5 + ............ + 17)

Formula :

1 + 3 + 5 + ............ + l = [(l + 1)/2]²

Then,

= [(325 + 1)/2]² - [(17 + 1)/2]²

= (326/2)² - (18/2)²

= 163² - 9²

= 26569 - 81

= 26488

10. Answer :

2 + 6 + 12 + 20 + ............ + 650 :

= (1 + 1) + (4 + 2) + (9 + 3) + (16 + 4) + ............ + (625 + 25)

= (1² + 1) + (2² + 2) + (3² + 3) + (4² + 4) + ......... + (25² + 25)

= (1² + 2² + 3² + 4²......... + 25²) + (1 + 2 + 3 + 4 + ......... + 25)

= [25(25 + 1)(2(25) + 1)]/6 + 25(25 + 1)/2

= [25(26)(51)]/6 + 25(26)/2

= 25(13)(17) + 25(13)

= 5525 + 325

= 5850

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