Further Mathematics

1. 1.

   Use partial fractions to find sum from r = 1 to 10 of 1/[r(r + 1)].

   [3 marks] · no calculator
2. 2.

   A geometric series has terms 12, -6, 3, ... . Find its sum to infinity and the exact error after four terms.

   [3 marks] · no calculator
3. 3.

   Prove by induction that sum from r = 1 to n of r^2 equals n(n + 1)(2n + 1)/6 for n >= 1.

   [4 marks] · no calculator
4. 4.

   Prove that 7^n - 1 is divisible by 6 for every integer n >= 1 using induction.

   [4 marks] · no calculator
5. 5.

   u1 = 1 and u(n + 1) = 2u(n) + 3. Conjecture a formula for u(n), prove it by induction, then find sum u(r) for r = 1,...,n.

   [3 marks] · no calculator
6. 6.

   Prove 2^n > n^2 for every integer n >= 5 by induction. Explain why starting at n = 4 cannot establish the strict inequality.

   [4 marks] · no calculator

Original, unofficial practice questions from SubjectScout — not official past-paper questions.