Section Check In 4.01 Proof – Questions 2 1. Prove by induction that is divisible by for . 2.* Prove by mathematical induction that is true for all . 3. Let the matrix . Prove by induction that for . 4. A sequence is given by , . Prove by induction that . 5.* Prove that the nth derivative of isfor . f 6.* Let be the function given by with . Using composition of functions and mathematical induction prove that for all . 3, 5 7 7. is thought to be divisible by and for all . (i) Show that this is not true. 3, 5 7 (ii) State which of and do divide into for all , and prove this by induction. 8.* Prove by mathematical induction that is true for all . 9. Prove that for positive integer n. 10.* The number of members of an online forum after days, , is modelled by where and and n a k are constants. 1 Given that the forum has one member on day and welcomes four new members per day, prove by induction that on the th day the number of members can be modelled by . Extension Let be an integer. Show that . [You may use the result that is an increasing function and .] Worked solutions 1. *Consider divisible by . Therefore true for . *Assume true for Let Then is assumed divisible by and for some . *Consider
Section Check In – 4.01 Proof Questions 1. Prove by induction that is divisible by 2 for . 2.* Prove by mathematical induction that is true for all . 3. Let the matrix . Prove by induction that for . 4. A sequence is given by , . Prove by induction that . 5.* Prove that the nth derivative of isfor . 6.* Let f be the function given by with . Using composition of functions and mathematical induction prove that for all . 7. is thought to be divisible by 3, 5 and 7 for all . (i) Show that this is not true. (ii) State which of 3, 5 and 7 do divide into for all , and prove this by induction. 8.* Prove by mathematical induction that is true for all . 9. Prove that for positive integer n. 10.* The number of members of an online forum after n days, , is modelled by where and a and k are constants. Given that the forum has one member on day 1 and welcomes four new members per day, prove by induction that on the th day the number of members can be modelled by . Extension Let be an integer. Show that . [You may use the result that is an increasing function and .] Worked solutions 1. *Consider divisible by . Therefore true for . *Assume true for Let Then is assumed divisible by and for some . *Consider
411688-section-check-in-4.01-proof.docx