4.4 Summing series by the method of differences Quiz: Pearson Edexcel Further Maths, Unit 4

The 20 questions

1. What is the main idea of the method of differences for summing a series?

   • Write each term as f(r) - f(r + 1) so that most terms cancel in the sum
   • Convert the series into an integral and evaluate the area under a curve
   • Compare each term with a geometric series to test for divergence
   • Write each term as a product of two equal factors so the sum factorises

2. Which partial fraction identity is used to sum the series of 1/(r(r + 1))?

   • 1/(r(r + 1)) = 1/r - 1/(r + 1)
   • 1/(r(r + 1)) = r/(r + 1)
   • 1/(r(r + 1)) = 1/r + 1/(r + 1)
   • 1/(r(r + 1)) = 1/(r + 1) - 1/r + 1

3. What is the sum to infinity of the series 1/(r(r + 1)) from r = 1?

   • 1/2
   • 0
   • infinity
   • 1

4. What is the value of the sum of 1/(r(r + 1)) from r = 1 to 9?

   • 9/10
   • 1
   • 1/10
   • 9/11

5. What is the sum of 1/((r + 1)(r + 2)) from r = 1 to 4?

   • 1/4
   • 1/3
   • 2/5
   • 1/2

6. What is the sum of 1/(r(r + 1)(r + 2)) from r = 1 to 2?

   • 1/12
   • 5/24
   • 1/6
   • 1/4

7. What is the value of the sum of r x r! from r = 1 to 3?

   • 23
   • 24
   • 22
   • 18

8. Which partial fraction form is correct for 1/(r(r + 2))?

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

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

   • 1/3
   • 5/12
   • 7/24
   • 11/24

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

    • 1/2
    • 1
    • 3/4
    • 1/3

11. What is the sum from r = 1 to n of (r^2 - (r - 1)^2)?

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

12. What is the sum from r = 1 to n of 1/(4r^2 - 1)?

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

13. What is the sum from r = 1 to n of ln((r + 1)/r)?

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

14. What is the sum from r = 1 to n of 1/(sqrt(r) + sqrt(r + 1))?

    • sqrt(n) - 1
    • sqrt(n + 1) - 1
    • sqrt(n + 1)
    • sqrt(n + 1) + 1

15. What is the sum from r = 1 to n of r/(r + 1)!?

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

16. What is the sum from r = 10 to 20 of 1/(r(r + 1))?

    • 1/10
    • 11/210
    • 1/21
    • 10/21

17. As n tends to infinity, the sum from r = 1 to n of 1/(r(r + 1)(r + 2)) tends to which value?

    • infinity
    • 1/6
    • 1/2
    • 1/4

18. What is the sum from r = 1 to n of (r^3 - (r - 1)^3)?

    • n^3
    • n^2
    • n^3 - 1
    • 3n^2 - 3n + 1

19. What is the sum from r = 1 to n of 2^(r - 1)?

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

20. The sum from r = 1 to n of 1/(r(r + 1)) equals 99/100. What is n?

    • 100
    • 98
    • 99
    • 50

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