D3-D4 Sums of series and the method of differences Quiz: AQA Further Maths, Unit 2

The 20 questions

1. What is the sum of the first n positive integers?

   • n(n + 1)(2n + 1)/6
   • n^2
   • n(n + 1)/2
   • n(n - 1)/2

2. What is the sum of the first n squares?

   • n(n + 1)(2n + 1)/6
   • n^3/3, which approximates the sum of the squares for large n
   • n(n + 1)/2, which is the formula for the sum of the integers
   • n^2(n + 1)^2/4, which is the formula for the sum of the cubes

3. What is the sum of the first n cubes?

   • n^3
   • (n(n + 1)/2)^2
   • n(n + 1)(2n + 1)/6
   • n^2(n + 1)/2

4. What is the sum of the squares 1^2 + 2^2 + ... + 10^2?

   • 3025
   • 385
   • 55
   • 220

5. What is the sum of the cubes 1^3 + 2^3 + ... + 5^3?

   • 225
   • 125
   • 275
   • 15

6. What is the sum of 1/(r(r + 1)) for r from 1 to n?

   • 1/(n + 1)
   • n/(n + 2)
   • n/(n + 1)
   • (n + 1)/n

7. What is the sum of 1/((2r - 1)(2r + 1)) for r from 1 to n?

   • n/(2n - 1)
   • n/(2n + 1)
   • 2n/(2n + 1)
   • 1/(2n + 1)

8. What is the sum of r(r + 1) for r from 1 to 10?

   • 55
   • 440
   • 495
   • 385

9. What is the sum of r(r + 1)(r + 2) for r from 1 to 2?

   • 60
   • 30
   • 24
   • 36

10. What is the limit of the sum of 1/(r(r + 2)) over r from 1 to infinity?

    • 1/2
    • 1/4
    • 1
    • 3/4

11. What is the sum of r^3 - (r - 1)^3 for r from 2 to 5?

    • 125
    • 124
    • 100
    • 99

12. Simplify the sum of r from 1 to 2n minus the sum of r from 1 to n.

    • n(n + 1)/2
    • n(3n + 1)/2
    • 2n(2n + 1)
    • n(3n - 1)/2

13. What is the limit of the sum of 1/(r(r + 1)(r + 2)) from r = 1 to n as n tends to infinity?

    • 1/3
    • 1/4
    • 1/6
    • 1/2

14. Which expression is the sum of (r^2 - 2r) for r from 1 to n?

    • n(n + 1)(2n - 1)/6
    • n(n + 1)(2n + 5)/6
    • n(n - 1)(2n - 5)/6
    • n(n + 1)(2n - 5)/6

15. What is the smallest n for which the sum of 1/(r(r + 1)) from 1 to n exceeds 0.9?

    • 11
    • 10
    • 8
    • 9

16. What is the sum of (2r + 1) for r from 1 to 3?

    • 9
    • 15
    • 12
    • 18

17. For the telescoping sum of 2/(r(r + 1)) from r = 1 to n, what is the limit as n tends to infinity?

    • 2
    • 1/2
    • infinity
    • 1

18. What is the sum of r^2 plus the sum of r for r from 1 to 4?

    • 50
    • 36
    • 40
    • 30

19. What is the sum of the first 20 odd numbers 1 + 3 + ... + 39?

    • 380
    • 390
    • 400
    • 420

20. Using the method of differences, what is the sum from r = 1 to n of (1/r - 1/(r + 1))?

    • 1 - 1/(n + 1)
    • 1 - 1/n
    • 1 + 1/(n + 1)
    • 1/(n + 1)

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