SECTION 12.5 Mathematical Induction 897 9. ( ) + + + + = − − 1 4 4 4 1 3 4 1 n n 2 1 10. ( ) + + + + = − − 1 5 5 5 1 4 5 1 n n 2 1 11. ( ) ⋅ + ⋅ + ⋅ + + + = + n n n n 1 1 2 1 2 3 1 3 4 1 1 1 12. ( )( ) ⋅ + ⋅ + ⋅ + + − + = + n n n n 1 1 3 1 3 5 1 5 7 1 2 1 2 1 2 1 13. ( )( ) + + + + = + + n n n n 1 2 3 1 6 1 2 1 2 2 2 2 14. ( ) + + + + = + n n n 1 2 3 1 4 1 3 3 3 3 2 2 15. ( ) ( ) + + + + − = − n n n 4 3 2 5 1 2 9 16. ( ) ( ) − − − − − + = − + n n n 2 3 4 1 1 2 3 17. ( ) ( )( ) ⋅+⋅+⋅++ += + + n n n n n 1 2 2 3 3 4 1 1 3 1 2 18. ( )( ) ( )( ) ⋅+⋅+⋅++ − = + − n n n n n 1 2 3 4 5 6 2 1 2 1 3 1 4 1 19. + n n 2 is divisible by 2. 20. + n n2 3 is divisible by 3. 21. − + n n 2 2 is divisible by 2. 22. ( )( ) + + n n n 1 2 is divisible by 6. Applications and Extensions In Problems 23–27, prove each statement. 23. If > x 1, then > x 1. n 24. If < < x 0 1, then < < x 0 1. n 25. −a b is a factor of − a b . n n [Hint: ( ) ( ) − = − + − + + a b a a b b a b k k k k k 1 1 ] 26. +a b is a factor of + + + a b . n n 2 1 2 1 27. ( ) + ≥ + a na 1 1 , n for > a 0 28. Show that the statement “ − + n n 41 2 is a prime number” is true for = n 1 but is not true for = n 41. 29. Show that the formula + + + + = + + n n n 2 4 6 2 2 2 obeys Condition II of the Principle of Mathematical Induction.That is, show that if the formula is true for some k, it is also true for +k 1. Then show that the formula is false for = n 1 (or for any other choice of n). 30. Use mathematical induction to prove that if ≠ r 1, then + + + + = − − − a ar ar ar a r r 1 1 n n 2 1 31. Use mathematical induction to prove that ( ) ( ) ( ) [ ] ( ) + + + + + + + − = + − a a d a d a n d na d n n 2 1 1 2 32. Extended Principle of Mathematical Induction The Extended Principle of Mathematical Induction states that if Conditions I and II hold, that is, (I) A statement is true for a natural number j. (II) If the statement is true for some natural number ≥ k j, then it is also true for the next natural number +k 1. then the statement is true for all natural numbers ≥ j. Use the Extended Principle of Mathematical Induction to show that the number of diagonals in a convex polygon of n sides is ( ) − n n 1 2 3 . [Hint: Begin by showing that the result is true when = n 4 (Condition I).] 33. Geometry Use the Extended Principle of Mathematical Induction to show that the sum of the interior angles of a convex polygon of n sides equals ( ) − ⋅ ° n 2 180 . 34. Challenge Problem Use the Principle of Mathematical Induction to prove that n n n n 5 8 2 3 4 1 2 8 1 4 n − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ for all natural numbers n. 35. Challenge Problem Paper Creases If a sheet of paper is folded in half by folding the top edge down to the bottom edge, one crease will result. If the folded paper is folded in the same manner, the result is three creases. With each fold, the number of creases can be defined recursively by = = + + c c c 1, 2 1. n n 1 1 (a) Find the number of creases for = n 3 and = n 4 folds. (b) Use the given information and your results from part (a) to find a formula for the number of creases after n folds, c ,n in terms of the number of folds alone. (c) Use the Principle of Mathematical Induction to prove that the formula found in part (b) is correct for all natural numbers. (d) Tosa Tengujo is reportedly the world’s thinnest paper with a thickness of 0.02 mm. If a piece of this paper could be folded 25 times, how tall would the stack be? Explaining Concepts 36. How would you explain the Principle of Mathematical Induction to a friend?
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