 # Proofs By Induction

## Question

Show transcribed text

## Transcribed Text

1. Prove by induction that 1 · 2 + 2 · 3 + 3 · 4 + · · · + n(n + 1) = n(n + 1)(n + 2) 3 2. Prove by induction that for all positive integers n 6|2n 3 + 3n 2 + n. 3. Prove by induction that for all positive integers n that Xn i=1 i(i!) = (n + 1)! − 1, 4. Find an expression for 1 − 3 + 5 − 7 + 9 − 11 + · · · + (−1)n−1 (2n − 1) and prove using induction that your expression is correct. 5. Let {f0, f1, f2, . . . } be the Fibonacci numbers. So f0 = 1 and f1 = 1 and fi = fi−1 + fi−2. (So f2 = 2, f3 = 3 and f4 = 5 and so on.) (a) Prove by induction that f0 + f1 + · · · + fk = fk+2 − 1. (b) Prove by induction that Xn i=0 f 2 i = fnfn+1. (c) Prove by strong induction that fn+5 ≡ 3fn (mod 5) 6. Consider a sequence {x1, x2, x3, x4, x5, ....} where x1 = 3 and x2 = 7 and for all other k we have that xk = 5xk−1 − 6xk−2. Using strong induction prove that for all n that xn = 2n + 3n−1

## Solution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

\$30.00 for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

### Find A Tutor

View available Advanced Math Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

SUBMIT YOUR HOMEWORK
We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

• 1
• 2
• 3
Live Chats