Induction Proofs

[discretec]

Instructions: Use mathematical induction to prove the truth of the following for all n>=1.
Problem: 8^n-3^n is divisible by 5.
n=1
8^1-3^1=5
Assume n=k
5 divides 8^k-3^k
n=k+1
8^k+1-3^k+1

I am stuck on how to go about solving this one. Any help is appreciated.

 

[Deleted member 4993]

Instructions: Use mathematical induction to prove the truth of the following for all n>=1.
Problem: 8^n-3^n is divisible by 5.
n=1
8^1-3^1=5
Assume n=k
5 divides 8^k-3^k
n=k+1
8^k+1-3^k+1

I am stuck on how to go about solving this one. Any help is appreciated.

Hint:

\(\displaystyle 8^{k+1} - 3^{k+1} = 8*(8^k - 3^k) + 5 * 3^k\)

 

[discretec]

Instructions: Use mathematical induction to prove the truth of the following for all n>=1.
Problem: 8^n-3^n is divisible by 5.
n=1
8^1-3^1=5
Assume n=k
5 divides 8^k-3^k
n=k+1
8^k+1-3^k+1

I am stuck on how to go about solving this one. Any help is appreciated.

Hint:

\(\displaystyle 8^{k+1} - 3^{k+1} = 8*(8^k - 3^k) + 5 * 3^k\)

I have the problem worked out and the final solution, but I am having trouble understanding how this problem is worked. Could you please explain this problem?
For all n >= 1, 8^n – 3^n is divisible by 5.
Let n = 1.
8^n – 3^n = 8^1 – 3^1 = 8 – 3 = 5
Assume n=k
5 divides 8^k – 3^k
n=k + 1.
8^k+1 – 3^k+1 = 8^k+1 – 3×8^k + 3×8^k – 3^k+1
= 8^k(8 – 3) + 3(8^k – 3^k) = 8^k(5) + 3(8^k – 3^k)
The first term in 8^k(5) + 3(8^k – 3^k) has 5 as a factor, and the second term is divisible by 5 from the hypothesis. Since 5 can be factored from these terms, 8^k(5) + 3(8^k – 3^k) = 8^k+1 – 3^k+1, must be divisible by 5.
Thank you for your help

 

[Deleted member 4993]

I have the problem worked out and the final solution, but I am having trouble understanding how this problem is worked. Could you please explain this problem?
For all n >= 1, 8^n – 3^n is divisible by 5.
Let n = 1.
8^n – 3^n = 8^1 – 3^1 = 8 – 3 = 5
Assume n=k
5 divides 8^k – 3^k
n=k + 1.
8^k+1 – 3^k+1 = 8^k+1 – 3×8^k + 3×8^k – 3^k+1
= 8^k(8 – 3) + 3(8^k – 3^k) = 8^k(5) + 3(8^k – 3^k)
The first term in 8^k(5) + 3(8^k – 3^k) has 5 as a factor, and the second term is divisible by 5 from the hypothesis. Since 5 can be factored from these terms, 8^k(5) + 3(8^k – 3^k) = 8^k+1 – 3^k+1, must be divisible by 5.
Thank you for your help

Where in the solution process - you are lost?

 

[discretec]

I have the problem worked out and the final solution, but I am having trouble understanding how this problem is worked. Could you please explain this problem?
For all n >= 1, 8^n – 3^n is divisible by 5.
Let n = 1.
8^n – 3^n = 8^1 – 3^1 = 8 – 3 = 5
Assume n=k
5 divides 8^k – 3^k
n=k + 1.
8^k+1 – 3^k+1 = 8^k+1 – 3×8^k + 3×8^k – 3^k+1
= 8^k(8 – 3) + 3(8^k – 3^k) = 8^k(5) + 3(8^k – 3^k)
The first term in 8^k(5) + 3(8^k – 3^k) has 5 as a factor, and the second term is divisible by 5 from the hypothesis. Since 5 can be factored from these terms, 8^k(5) + 3(8^k – 3^k) = 8^k+1 – 3^k+1, must be divisible by 5.
Thank you for your help

Where in the solution process - you are lost?

I am lost after the hypothesis when n=k+1. I do not understand how this is figured
n=k + 1.
8^k+1 – 3^k+1 = 8^k+1 – 3×8^k + 3×8^k – 3^k+1
= 8^k(8 – 3) + 3(8^k – 3^k) = 8^k(5) + 3(8^k – 3^k)
The first term in 8^k(5) + 3(8^k – 3^k) has 5 as a factor, and the second term is divisible by 5 from the hypothesis. Since 5 can be factored from these terms, 8^k(5) + 3(8^k – 3^k) = 8^k+1 – 3^k+1, must be divisible by 5.

 

[Deleted member 4993]

II am lost after the hypothesis when n=k+1. I do not understand how this is figured
n=k + 1.
8^k+1 – 3^k+1 = 8^k+1 – 3×8^k + 3×8^k – 3^k+1 <<< Those two cancel out
= 8^k(8 – 3) + 3(8^k – 3^k) = 8^k(5) + 3(8^k – 3^k)
The first term in 8^k(5) + 3(8^k – 3^k) has 5 as a factor, and the second term is divisible by 5 from the hypothesis. Since 5 can be factored from these terms, 8^k(5) + 3(8^k – 3^k) = 8^k+1 – 3^k+1, must be divisible by 5.

 

[discretec]

Thank you. I appreciate your help.