OCR Maths FP1. Topic Questions from Papers. Summation of Series. Answers

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1 OCR Maths FP Topic Questions from Papers Summation of Series Answers PhysicsAndMathsTutor.com

2 . Σr +Σr + Σ Σr = n(n +)(n +) Σr = n(n +) Σ = n PhysicsAndMathsTutor.com Consider the sum of three separate terms Correct formula stated Correct formula stated Correct term seen. n(n +n +) Correct algebraic processes including factorisation and simplification (Q, June 005) 5. (i) (r +) r(r +) (r +)(r +) (r + )(r +) Show correct process for subtracting fractions EITHER +... n + n + n n + Express terms as differences using (i) At least first two and last term correct n + n + Show or imply that pairs of terms cancel Obtain correct answer in any form OR M B ft 7 State that Σ n ur r = Each term correct = f(n +) f() Obtain value from their sum to n terms. (i) Circle B Sketch(s) showing correct features, each mark (Q5, June 005) 5. 8Σr Σr +Σ r 8Σr =n (n +) Consider the sum of three separate terms Correct formula stated or used a.e.f. Σr = n(n +)(n +) Correct formula stated or used a.e.f. Σ r = n(n +) Correct term seen n (n +) AG Attempt to factorise or expand and simplify (Q5, Jan 00)

3 9. (i) r + r r(r +) r(r +) AG Show correct process for subtracting fractions Express terms as differences using (i) Express st (or last ) terms so that cancelling occurs Obtain + n + n + 5 Obtain n +, n + Obtain correct answer in any form (a) Bft Obtain value from their sum to n terms (b) n + + n + ft Using (a) or method of differences again [ n is a method error ] Obtain answer in any form 0. (i) 0 (Q9, Jan 00). 5 Σ r + Σ r Consider the sum as two separate parts Σr = n(n +)(n +) Correct formula stated Σ r = n (n +) Correct formula stated n(n +)(n +)(n +) 5 Attempt to factorise and simplify or expand both expressions or complete verification 5 5. (i) -7i B Real part correct (Q, June 00)

4 9. (i) Show that terms cancel in pairs Attempt to expand and simplify (n +) n(n +) n B B Correct Σr stated Σ = n Consider sum of three separate terms on RHS Required sum is LHS two terms Correct unsimplified expression n(n +)(n +) 0 (Q9, June 00). 7 Expand to obtain r r n n + ) n ( n + ) ( n ( n )( n + )( n + ) Consider difference of two standard results Obtain correct unfactorised answer Attempt to factorise Obtain factor of n ( n + ) Obtain correct answer. (i) B Circle (Q, Jan 007)

5 8. 8 (i) Factor of r! or (r + )! seen Factor of (r + ) found (r + ) r! Express terms as differences using (i) At least st two and last term correct (n + )!! Show that pairs of terms cancel Bft Obtain correct answer in any form 8 Convincing statement for nonconverging, ft their 9. For at least two correct images (Q8, Jan 007) 9 Σr Σr + Σ 5 Consider the sum of three separate terms Σr = n(n +)(n +) Correct formula stated Σr = n(n +) Correct formula stated Σ = n n Correct term seen Attempt to simplify B Transpose leading diagonal and (Q, negate June other 007) 0 5 (i) r(r +) n + S = n + B Bft c.a.o. 7 Show correct process to obtain given result Express terms as differences using (i) Show that terms cancel Obtain correct answer, must be n not any other letter State correct value of sum to infinity Ft their Use sum to infinity their Obtain correct answer a.e.f. (Q5, June 007)

6 a n( n + )(n + ) + bn Each column correct Consider sum as two separate parts Correct answer a.e.f. a = b = Compare co-efficients Obtain correct answers Use given substitution (Q, Jan 008) 0 (i) Attempt to combine fractions + n + n + Express at least first terms using (i) All terms correct Express at least last terms using (i) All terms correct in terms of n Show that correct terms cancel Obtain unsimplified correct answer 5 Bft Obtain correct answer from their + + = (iv) 7 N + N 0 Bft Their their 7N 9N = 0 N = Attempt to clear fractions & solve equation, Obtain correct simplified equation Obtain only the correct answer (Q0, Jan 008) (i) r ( r + )! ( n + )! Common denominator of (r + )! or r!(r + )! Express terms as differences using (i) At least st two and last term correct Show pairs cancelling Correct answer a.e.f. B Establish result is true, for n = ( or or ) (Q, June 008)

7 5 Express as difference of two series Use standard results n ( n+ ) n( n+ )(n+ ) Correct unsimplified answer Attempt to factorise At least factor of n(n + ) ( )( )( ) nn + n + n Obtain correct answer (i) i B Conjugate stated (Q5, June 008) 5 Express as sum of terms n ( n + ) + n ( n + )(n + ) + n ( n + ) correct unsimplified terms rd correct unsimplified term n ( n + ) ( n + ) Attempt to factorise ft Two factors found, ft their quartic Correct final answer a.e.f. B State or use correct result (Q, Jan 009) 9 (i) 0 Use correct denominator + n n + Express terms as differences using (i) Do this for at least st terms First terms all correct Last terms all correct ( in terms or n or r) Show pairs cancelling Obtain correct answer, a.e.f.( in terms of n) Bft 9 Given answer deduced correctly, ft their (Q9, Jan 009) = B State correct value of S 50 or S 00 Subtract S 50 S 00 ( or S 0 or S 99 ) Obtain correct exact answer. a +5b =, a + b = Obtain a pair of simultaneous (Q, June 009)

8 7. 8 (i) Show that terms cancel in pairs Attempt to expand and simplify B B Correct r stated n ( n ) n ( n )(n ) n ( n ) n * *D Consider sum of separate terms on RHS Required sum is LHS terms Correct unsimplified expression n r n ( n ) r 0 8. (i) B Find coordinates (0, 0) (, (Q7, ) June (, ) 009) 9 Express as sum of three series Use standard results n ( n + ) n ( n + )(n + ) n ( n + ) Obtain correct unsimplified answer Attempt to factorise Obtain at least factor of n ( n + ) n ( n + )( n + )(n 7) Obtain fully factorised correct answer (Q, Jan 00) 0 7 (i) B Express at least st two and last term using (i) All terms correct Show that correct terms cancel ( n + ) Obtain correct answer, in terms of n B Sum to infinity seen or implied B Obtain correct answer S.C. -¾ scores B 7 (Q7, Jan 00)

9 Either n ( n + )(n + ) n ( n + ) + n Express as a sum of terms Use standard sum results Correct unsimplified answer n (n )(n + ) Or n r = r n r = r n (n + )(n + ) n ( n + )(n + ) n (n )(n + ) Attempt to factorise Obtain at least factor of n and a quadratic Obtain correct answer a.e.f. Express as difference of Use standard result Correct unsimplified answer Attempt to factorise Obtain at least factor of n Obtain correct answer r series (Q, June 00) 8 (i) Attempt to rationalise denominator or cross multiply ( n + + n + ) Express terms as differences using (i) Attempt this for at least st three terms st three terms all correct Last two terms all correct Show pairs cancelling Obtain correct answer, in terms of n B 9 Sensible statement for divergence (Q8, June 00)

10 Either B Correct value for r stated or used Express as sum of two series a bn n ( n + ) + ( n + ) Obtain correct unsimplified answer Compare coefficients or substitute values for n a = b = Obtain correct answers Or Use values for n a + b = 0 a + b = Obtain correct equations Solve simultaneous equations a = b = Obtain correct answers 5 B (A - ) - = A seen or implied (Q, Jan 0) 0 (i) Use correct denominator Express terms as differences using (i) Do this for at least terms First terms all correct Last terms all correct + n + n + Show relevant cancelling Obtain correct answer a.e.f Bft S stated or start at n + as in S n + n - their or show correct cancelling + ( n + )( n + ) (Q0, Jan 0) 5 Express as sum of two series n (n )(n ) n Each term correct a.e.f. n ( n ) Attempt to factorise A Completely correct answer, ( if one factor not found ) (Q, June 0)

11 7 (i) B Express at least st two and last two terms using (i) st two terms correct Last two terms correct Show that correct terms cancel n ( n ) 5 Obtain correct answer, a.e.f. in terms of n Bft Sum to infinity stated or implied or start at 000 as in S their with n = 999 or 000 or show correct cancelling Obtain correct answer, a.e.f. ( condone 0.00 ) 9 (Q7, June 0) 7 Express as difference of two series D Use standard series results n ( n ) n ( n ) Obtain correct unsimplified answer Attempt to factorise At least factor of n( n + ) From their unsimplified answer nn ( )( n )( n ) Obtain correct answer [] (Q, Jan 0) 8 8 (i) Combine with a common denominator [] 8 Express terms using (i) At least st two and last two correct n Show terms cancelling n Obtain correct answer, in terms of n [] 8 n n B lim n n n BFT This value [] (Q8, Jan 0)

12 9 Express as sum of series Use standard series results, at least correct Two terms correct nn ( )(n ) nn ( ) n Third term correct Obtain factor of n nn ( ) A Obtain correct answer c.a.o. [7] Allow for ( n ) (Q, June 0) 0 8 (i) B Show given answer correctly [] 8 Express terms as differences using (i) Attempt this for at least first terms First terms all correct Last terms correct Show terms cancelling Obtain correct answer, must be in terms of n n n [] 8 Bft State or use correct sum to infinity B Their sum to infinity their = Attempt to solve correct equation N = Obtain only N = [] (Q8, June 0) nn ( )( n ) n * Attempt to expand (r )(r +) D Use standard result for r² Obtain correct unsimplified answer [] 5 n( n )( n ) D Attempt to factorise A [] (i) B Allow. Obtain completely correct answer Allow if one bracket still contains a common factor (Q, Jan 0) [] 8 (i) Obtain correct numerator from addition or partial fractions [] 8 Express at least three relevent terms using (i) st three terms correct n Last two terms correct ( n )( n ) Show correct cancelling [5] 8 Sum to - st term or start process at r = Obtain correct answer [] (Q8, Jan 0)

13 n ( n ) n( n )(n ) n( n ) n ( n ) A 5 [] Express as sum of three series Obtain correct (unsimplified ) terms Obtain correct rd (unsimplified) term Attempt to factorise, at least factor of n Obtain correct answer, if not fully factorised (Q5, June 0) 9 (i) Use correct denominator or partial fractions Obtain given answer convincingly [] Express at least st two and last term using (i) All terms correct Show correct terms cancelling n Obtain correct unsimplified answer [] Include and combine their sum as a single fraction Obtain given answer (Q9, June 0)

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

OCR Maths FP1. Topic Questions from Papers. Proof by Induction. Answers OCR Maths FP Topic Questions from Papers Proof by Induction Answers PhysicsAndMathsTutor.com (iv) Explicit check for n = or n = M k = k ( k ) 0. Induction hypothesis that result is true for M k Attempt