A-LevelFurther MathematicsProof by induction9231/12 · May/June 2013 · Q25 Marks
Prove by mathematical induction that 5^(2n) – 1 is divisible by 8 for every positive integer n.
- (a)Prove by mathematical induction that 5^(2n) – 1 is divisible by 8 for every positive integer n.[5]
Mark scheme
Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.
- B1 Proves base case.
- B1 Pn: 5^(2n) – 1 is divisible by 8. 5² – 1 = 24 = 3 × 8 ⇒ P₁ is true
- M1 States inductive hypothesis.
- Assume Pk is true: 5^(2k) – 1 = 8λ for some k.
- 5^(2k+2) – 1 = 25.5^(2k) – 1 = 24.5^(2k) + 5^(2k) – 1 = 3 × 8.5^(2k) + 8λ
- A1 Proves inductive step.
- ∴ Pk ⇒ Pk+1
- A1 States conclusion.
- (Since P₁ is true and Pk ⇒ Pk+1).. Pn is for every positive integer n (by PMI).
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