Question: Use induction to prove that (f(n)=8^{n}-1) is divisible by 7 for all natural numbers (n). For the induction step, show that [ 8^{k+1}-1=7 cdot 8^{k}+left(8^{k}-1ight)

Use induction to prove that $f(n)=8^{n}-1$ is divisible by 7 for all natural numbers $n$. For the induction step, show that

\[
8^{k+1}-1=7 \cdot 8^{k}+\left(8^{k}-1ight)
\]

Step by Step Solution

★★★★★

3.32 Rating (155 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

First a brief explanation of induction Mathematical induction is a method used to prove a statement ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Calculus 4th Questions!

In this question we study the recursively defined functions f, g and h given by the following defining equations f(0) = 1 base case 0, f(1) = 0 base case 1, and f(n) = n f(n 1) + f(n 2)^2...

Use induction to prove that $5^{2 n}-4^{n}$ is divisible by 7 for all natural numbers $n$.

What precedent would be MOST useful to you if you were the creator of a website called csufsucks.com and a federal judge issued an injunction ordering you to take it down and stop future posts on the...

Please answer the 6th question. I. (2 points) Let F, be the Fibonacci numbers, where Fo= l, F,-I, F2-2, F,-3, F. = 5 . Prove -2 F1 = FN-2, where N> 2 using induction. (Lipschutz, HW 3) 2 C2 points)...

Please answer the 4th question. I. (2 points) Let F, be the Fibonacci numbers, where Fo= l, F,-I, F2-2, F,-3, F. = 5 . Prove -2 F1 = FN-2, where N> 2 using induction. (Lipschutz, HW 3) 2 C2 points)...

Please answer the second question. I. (2 points) Let F, be the Fibonacci numbers, where Fo= l, F,-I, F2-2, F,-3, F. = 5 . Prove -2 F1 = FN-2, where N> 2 using induction. (Lipschutz, HW 3) 2 C2...

Please answer the third question. I. (2 points) Let F, be the Fibonacci numbers, where Fo= l, F,-I, F2-2, F,-3, F. = 5 . Prove -2 F1 = FN-2, where N> 2 using induction. (Lipschutz, HW 3) 2 C2 points)...

Please answer the 8th question. I. (2 points) Let F, be the Fibonacci numbers, where Fo= l, F,-I, F2-2, F,-3, F. = 5 . Prove -2 F1 = FN-2, where N> 2 using induction. (Lipschutz, HW 3) 2 C2 points)...

Please answer the 10th question. I. (2 points) Let F, be the Fibonacci numbers, where Fo= l, F,-I, F2-2, F,-3, F. = 5 . Prove -2 F1 = FN-2, where N> 2 using induction. (Lipschutz, HW 3) 2 C2 points)...

CONCEPTS 9. 1' + 2 + 3 3 + . . . + 1) = n (n + 1)2 1. Mathematical induction is a method of proving that a statement 4 P(n) is true for all numbers n. In Step I we prove 10. 13 + 3 + 53 + . .. + (2n...

Is biological diversity a renewable or nonrenewable resource? Could it be seen both ways? Explain.

Imagine you are part of a team developing a microservice - based online bookstore called "BookBarn". BookBarn aims to be a leading online destination for readers, authors, and publishers. It hosts a...

5. How were the antitrust laws used to deter union activities?

Determine whether the series converges or diverges. 2 sin n=1

Determine whether the series converges or diverges. 1+1/n n=1 n

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 5 + n -1

Find the monthly house payments necessary to amortize an 8.4% loan of $285,500 over 30 years. The payment size is $. (Round to the nearest cent.)

The following items are taken from the financial statements of Harris Services for 2024: Accounts payable $18,500 Accounts receivable 4,510 Accumulated depreciation-equipment 4,860 Accumulated...

Please show the specific details of thecalculation Green FinTechnoloy Cop. launched its IPO last month, and just paid its annual dividend of $0.5 per share. As a rising star, it is expected to...