OCR Maths FP1. Topic Questions from Papers. Proof by Induction. Answers

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1 OCR Maths FP Topic Questions from Papers Proof by Induction Answers PhysicsAndMathsTutor.com

2 (iv) Explicit check for n = or n = M k = k ( k ) 0. Induction hypothesis that result is true for M k Attempt to multiply MM k or vice versa k + ( k + ) 0. Element ( k+ ) derived correctly All other elements correct Explicit statement of induction conclusion (Q9, June 00). = Show result true for n = or n(n +)(n +)+(n +) D Add next term to given sum formula, any letter OK Attempt to factorise or expand and simplify (n +)(n +){(n +)+} Correct expression obtained Specific statement of induction conclusion, with no errors seen. (Q, Jan 00). (i) Attempt at matrix multiplication A = 0 A = Correct A Correct A (ii) A n = (iii) n Sensible conjecture made State that conjecture is true for n = or Attempt to multiply A n and A or vice versa Obtain correct matrix Statement of induction conclusion 8. (i) (Q, June 00)

3 . (i) Correct expression for u n+ Attempt to expand and simplify u n+ u n = n + Obtain given answer correctly (ii) State u = ( or u = 0 )and is divisible by State induction hypothesis true for u n Attempt to use result in (ii) Correct conclusion reached for u n+ 8 Clear,explicit statement of induction conclusion. (i) + = =0 State correct values (Q, Jan 00) ( =) n (n +) +(n +) (n +) (n +) (indep) Show result true for n = Add next term to given sum formula Attempt to factorise and simplify Correct expression obtained convincingly Specific statement of induction conclusion Consider the sum of three separate (Q, terms June 00) 8 (i) u =, u = 9, u = Obtain next terms All terms correct (ii) u n = n Sensible conjecture made 9 (iii) State that conjecture is true for n = or Find u n+ in terms of n Obtain ( n + ) Statement of Induction conclusion (Q8, Jan 008)

4 Establish result is true, for n = ( or or ) Attempt to multiply A and A n, or vice versa Correct process for matrix multiplication Obtain n +, 0 and Obtain ½( n + ) Statement of Induction conclusion, only if marks earned, but may be in body of working (Q, June 008) n n n + n 8 (i) Correct expression seen Attempt to factorise both terms in (i) (ii) Obtain correct expression Check that result is true for n = ( or ) Recognise that (i) is divisible by Deduce that u n+ is divisible by Clear statement of Induction conclusion 8 (i) Expand at least of the brackets (Q, Jan 009) 0. 9 i) u = u =9 0 Attempt to find next terms Obtain correct answers Show given result correctly (ii) u n = ( n )+ Expression involving a power of Obtain correct answer (iii) u n+ = (( n )+) u n + = ( n ) + ft 0 Verify result true when n = or n = Expression for u n + using recurrence relation Correct unsimplified answer Correct answer in correct form Statement of induction conclusion (Q0, June 009)

5 0 (i) M = 0 M = 0 Correct M seen Convincing attempt at matrix multiplication for M Obtain correct answer (ii) M n n = ft State correct form, consistent with (i) (iii) Correct attempt to multiply M & M k or v.v. Obtain element ( k + ) PhysicsAndMathsTutor.com Clear statement of induction step, from correct working Clear statement of induction conclusion, following their working (Q0, Jan 00) Establish result true for n = or n = Add next term to given sum formula Attempt to factorise or expand and simplify to correct expression Correct expression obtained Specific statement of induction conclusion (Q, June 00) * Establish result true for n = or * Use given result in recurrence relation in a relevant way * Obtain n + correctly dep Specific statement of induction conclusion (Q, Jan 0) Establish result true for n = or * Add next term to given sum formula D Combine with correct denominator Obtain correct expression convincingly Specific statement of induction conclusion, provided st marks earned (Q, June 0)

6 (i) Attempt at matrix multiplication Obtain M correctly Obtain given answer correctly [] (ii) n 0 n [] elements correct th element correct (iii) Show that their result is true for n = or Attempt to find M k. M or vice versa k 0 Obtain correct answer k Complete statement of induction conclusion Must have st marks [] (Q, Jan 0) Verify result true when n = * Add next term in series Dep Attempt to obtain correctly Show sufficient working to justify correct expression Clear statements of Induction processes, but st marks must all be earned. [] (Q, June 0) 0 (i),, 0 (ii) n 0 (iii) ( n ) x, Obtain correct values Justify given answer [] Fraction, in terms of n, with correct numerator or denominator Obtain correct answer a.e.f. [] ft Verify result true when n =, for their (ii), or n =, or Expression for u n + using recurrence relation in terms of n using their (ii) Correct unsimplified answer Correct answer in correct form Specific statement of induction conclusion, previous marks must be earned, n= [] must be verified (Q0, Jan 0) Establish result true for n = or n = Multiply M and M k, either order k k k or Obtain correct element [] Obtain other correct elements Obtain k convincingly Specific statement of induction conclusion, provided / earned so far and verified for n = (Q, June 0)