# soln3 - MATH 135 Algebra Solutions to Assignment 3 1(a Let...

This preview shows pages 1–2. Sign up to view the full content.

MATH 135 Algebra, Solutions to Assignment 3 1: (a) Let a 1 = 1 and a n +1 = 3 a n + 2 for n 1. Show that a n = 2 · 3 n - 1 - 1 for all n 1. Solution: We claim that a n = 2 · 3 n - 1 - 1 for all n 1. When n = 1 we have a n = a 1 = 1 and 2 · 3 n - 1 - 1 = 2 · 3 0 - 1 = 2 · 1 - 1 = 1 so the claim is true when n = 1. Let k 1 and suppose the claim is true when n = k , that is suppose that a k = 2 · 3 k - 1 - 1. Then when n = k + 1 we have a n = a k +1 = 3 a k + 2 = 3 ( 2 · 3 k - 1 - 1 ) + 2 = 2 · 3 k - 3 + 2 = 2 · 3 k - 1 = 2 · 3 n - 1 - 1 . Thus the claim is true when n = k + 1. By Mathematical Induction, a n = 2 · 3 n - 1 - 1 for all n 1. (b) Let a 1 = 3 and a n +1 = 2 a n - 1 for n 1. Find a closed form formula for a n . Solution: Using the given recursion formula, we ﬁnd that a 1 = 3, a 2 = 5, a 3 = 9, a 4 = 17 and a 5 = 33. Notice that a n = 2 n + 1 for n = 1 , 2 , 3 , 4 , 5. We claim that a n = 2 n + 1 for all n 1. When n = 1 the claim is true. Let k 1 and suppose that the claim is true when n = k , that is suppose that a k = 2 k + 1. Then when n = k + 1 we have a n = a k +1 = 2 · a k - 1 = 2(2 k + 1) - 1 = 2 k +1 + 2 - 1 = 2 k +1 + 1 = 2 n + 1 . Thus the claim is true when n = k + 1. By Mathematical Induction, we have a n = 2 n + 1 for all n 1. (c) Let a 1 = 2 and a n +1 = 5 a n - 4 a n for n 1. Show that 1 a n a n +1 4 for all n 1. Solution: We claim that 1 a n a n +1 4 for all n 1. We have a 1 = 2 and the recursion formula gives a 2 = 5 a 1 - 4 a 1 = 5 · 2 - 4 2 = 3, and so we do have 1 a 1 a 2 4 and so the claim is true when n = 1. Let k 1 and suppose the claim is true when n = k , that is suppose that 1 a k a k +1 4. We have 1 a k a k +1 4 = 1 1 a k 1 a k +1 1 4 = 4 4 a k 4 a k +1 1 = ⇒ - 4 ≤ - 4 a k ≤ - 4 a k +1 ≤ - 1 = 1 5 - 4 a k 5 - 4 a k +1 4 = 1 5 a k - 4 a k 5 a k +1 - 4 a k +1 4 = 1 a k +1 a k +2 4 . Thus the claim is true when n = k + 1. By Mathematical Induction, 1 a n a n +1 4 for all n 1. 2: (a) Let a 0 = 0 and a 1 = 1 and for n 2 let a n = a n - 1 +6 a n - 2 . Show that a n = 1 5 ( 3 n - ( - 2) n ) for all n 0. Solution: We claim that

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 5

soln3 - MATH 135 Algebra Solutions to Assignment 3 1(a Let...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online