{"id":1076,"date":"2020-03-06T03:48:42","date_gmt":"2020-03-06T03:48:42","guid":{"rendered":"http:\/\/ibalmaths.com\/?post_type=knowledgebase&p=1076"},"modified":"2020-03-06T10:20:05","modified_gmt":"2020-03-06T10:20:05","slug":"ibdp-past-year-exam-questions-mathematical-induction","status":"publish","type":"knowledgebase","link":"https:\/\/ibalmaths.com\/index.php\/ibdp-math-hl-2\/mathematical-induction\/ibdp-past-year-exam-questions-mathematical-induction\/","title":{"rendered":"IBDP Past Year Exam Questions – Mathematical Induction"},"content":{"rendered":"

Q1.\u00a0 \u00a0[M09.P1.TZ2] & [N18.P1]<\/strong><\/p>\r\n

Prove by mathematical induction\u00a0 \r\n∑<\/mo>r<\/mi>=<\/mo>1<\/mn><\/mrow>n<\/mi><\/munderover> <\/mo>r<\/mi>r<\/mi>!<\/mo><\/mrow><\/mfenced>=<\/mo>n<\/mi>+<\/mo>1<\/mn><\/mrow><\/mfenced>!<\/mo>–<\/mo>1<\/mn>,<\/mo> <\/mo>n<\/mi>∈<\/mo>ℤ<\/mi>+<\/mo><\/msup><\/math>\r\n \u00a0 .\u00a0 [8]<\/p>\r\n

Q2.\u00a0 \u00a0[N09.P1]<\/strong><\/p>\r\n

Using mathematical induction, prove that\u00a0 \r\n∑<\/mo>r<\/mi>=<\/mo>1<\/mn><\/mrow>n<\/mi><\/munderover> <\/mo>r<\/mi>+<\/mo>1<\/mn><\/mrow><\/mfenced>2<\/mn>r<\/mi>–<\/mo>1<\/mn><\/mrow><\/msup>=<\/mo>n<\/mi>2<\/mn>n<\/mi><\/msup> <\/mo>,<\/mo> <\/mo>n<\/mi>∈<\/mo>ℤ<\/mi>+<\/mo><\/msup><\/math>\r\n .\u00a0 [7]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 .\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/p>\r\n

Q3.\u00a0 \u00a0[M10.P1]<\/strong><\/p>\r\n

(a)\u00a0\u00a0<\/strong> Consider the following sequence of equations.<\/p>\r\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \r\n1<\/mn>×<\/mo>2<\/mn>=<\/mo>1<\/mn>3<\/mn><\/mfrac>1<\/mn>×<\/mo>2<\/mn>×<\/mo>3<\/mn><\/mrow><\/mfenced><\/mspace>1<\/mn>×<\/mo>2<\/mn>+<\/mo>2<\/mn>×<\/mo>3<\/mn>=<\/mo>1<\/mn>3<\/mn><\/mfrac>2<\/mn>×<\/mo>3<\/mn>×<\/mo>4<\/mn><\/mrow><\/mfenced><\/mspace>1<\/mn>×<\/mo>2<\/mn>+<\/mo>2<\/mn>×<\/mo>3<\/mn>+<\/mo>3<\/mn>×<\/mo>4<\/mn>=<\/mo>1<\/mn>3<\/mn><\/mfrac>3<\/mn>×<\/mo>4<\/mn>×<\/mo>5<\/mn><\/mrow><\/mfenced><\/mspace>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo><\/math>\r\n<\/p>\r\n

(i)\u00a0<\/strong> Formulate a conjecture for the\u00a0 \r\nn<\/mi>t<\/mi>h<\/mi><\/mrow><\/msup><\/math>\r\n equation in the sequence.<\/p>\r\n

(ii)\u00a0\u00a0<\/strong>Verify your conjecture for\u00a0 \r\nn<\/mi>=<\/mo>4<\/mn><\/math>\r\n .\u00a0 \u00a0[2]<\/p>\r\n

(b)<\/strong>\u00a0 \u00a0 \u00a0 \u00a0A sequence of numbers has the \r\nn<\/mi>t<\/mi>h<\/mi><\/mrow><\/msup><\/math>\r\n term given by\u00a0 \r\nu<\/mi>n<\/mi><\/msub>=<\/mo>2<\/mn>n<\/mi><\/msup>+<\/mo>3<\/mn>,<\/mo> <\/mo> <\/mo>n<\/mi>∈<\/mo>ℤ<\/mi>+<\/mo><\/msup><\/math>\r\n . Bill conjectures that all members of the sequence are prime numbers. Show that Bill\u2019s conjecture is false. [2]<\/p>\r\n

(c)<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Use mathematical induction to prove that\u00a0 \r\n5<\/mn>×<\/mo>7<\/mn>n<\/mi><\/msup>+<\/mo>1<\/mn><\/math>\r\n is divisible by 6 for all\u00a0 \r\nn<\/mi>∈<\/mo>ℤ<\/mi>+<\/mo><\/msup><\/math>\r\n .\u00a0 [6]<\/p>\r\n

Q4.\u00a0 \u00a0[M08.P1]<\/strong><\/p>\r\n

Use mathematical induction to prove that for \r\nn<\/mi>∈<\/mo>ℤ<\/mi>+<\/mo><\/msup><\/math>\r\n ,<\/p>\r\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \r\na<\/mi>+<\/mo>a<\/mi>r<\/mi>+<\/mo>a<\/mi>r<\/mi>2<\/mn><\/msup>+<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>+<\/mo>a<\/mi>r<\/mi>n<\/mi>–<\/mo>1<\/mn><\/mrow><\/msup>=<\/mo>a<\/mi>1<\/mn>–<\/mo>r<\/mi>n<\/mi><\/msup><\/mrow><\/mfenced><\/mrow>1<\/mn>–<\/mo>r<\/mi><\/mrow><\/mfrac><\/math>\r\n .\u00a0 \u00a0 \u00a0 \u00a0 \u00a0[7]<\/p>\r\n

Q5.\u00a0 \u00a0[M11.P2] & [M18.P1]<\/strong><\/p>\r\n

Prove by mathematical induction that, for \r\nn<\/mi>∈<\/mo>ℤ<\/mi>+<\/mo><\/msup><\/math>\r\n ,<\/p>\r\n

\r\n1<\/mn>+<\/mo>2<\/mn>1<\/mn>2<\/mn><\/mfrac><\/mfenced>+<\/mo>3<\/mn>1<\/mn>2<\/mn><\/mfrac><\/mfenced>2<\/mn><\/msup>+<\/mo>4<\/mn>1<\/mn>2<\/mn><\/mfrac><\/mfenced>3<\/mn><\/msup>+<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>.<\/mo>+<\/mo>n<\/mi>1<\/mn>2<\/mn><\/mfrac><\/mfenced>n<\/mi>–<\/mo>1<\/mn><\/mrow><\/msup>=<\/mo>4<\/mn>–<\/mo>n<\/mi>+<\/mo>2<\/mn><\/mrow>2<\/mn>n<\/mi>–<\/mo>1<\/mn><\/mrow><\/msup><\/mfrac><\/math>\r\n .\u00a0 \u00a0[8]<\/p>\r\n

Q6.\u00a0 \u00a0[M17.P1]<\/strong><\/p>\r\n

Use the method of mathematical induction to prove that \r\n4<\/mn>n<\/mi><\/msup>+<\/mo>15<\/mn>n<\/mi>–<\/mo>1<\/mn><\/math>\r\n is divisible by\u00a0 \r\n9<\/mn><\/math>\r\n for\u00a0 \r\nn<\/mi>∈<\/mo>ℤ<\/mi>+<\/mo><\/msup><\/math>\r\n .\u00a0 [6]<\/p>\r\n