Discrete Math
posted by Francesca .
Use mathematical induction to prove the truth of each of the following assertions for all n ≥1.
5^2n – 2^5n is divisible by 7
If n = 1, then 5^2(1)  2^5(1) = 7, which is divisible by 7. For the inductive case, assume k ≥ 1, and the result is true for n = k; that is 7  (5^2k + 2^5k). Use the assumption to prove n = k + 1, in other words, 5^(2(k + 1))  2^(5(k + 1)) is divisible by 7. Now,
5^(2(k + 1))  2^(5(k + 1))
= 5^(2k + 2)  2(5k + 5)
= 5^(2k) · 5^2  2^(5k) · 2^5
= 25 · 5^(2k)  32 · 2^(5k)
= IDK what to do from here. . .
Any suggestions? Thank you again!

Let's continue:
25 · 5^(2k)  32 · 2^(5k)
=25*(5^(2k)2^(5k) 7*2^(5k)
Now ask yourself:
A. Is (5^(2k)2^(5k) divisible by 7, and why?
B. Is 7*2^(5k) divisible by 7, and why?
Respond to this Question
Similar Questions

Math
Use mathematical induction to prove that 5^(n)  1 is divisible by four for all natural numbers n. Hint: if a number is divisible by 4, then it has a factor of 4. also, 1 = 5 +4 This is a take home test so I don't want the answer … 
Calculus
Use mathematical induction to prove that each proposition is valid for all positive integral values of n. 5^n + 3 is divisible by 4. 
Discrete Math
Use mathematical induction to prove the truth of each of the following assertions for all n ≥1. n³ + 5n is divisible by 6 I really do not understand this to much. This is what I have so far: n = 1, 1³  5(1) = 6, which is divisible … 
Math
Use mathematical induction to prove that 2^(3n)  3^n is divisible by 5 for all positive integers. ThankS! 
MATHS
prove by mathematical induction that 7^n+4^n+1 is divisible by 6 
Algebra
Prove by mathematical induction that 3^(3n+1) + 2^(n+1) is divisible by 5 
Mathematical induction. I'm stuck. So far I have..
For all integers n ≥ 1, prove the following statement using mathematical induction. 1+2^1 +2^2 +...+2^n = 2^(n+1) −1 Here's what I have so far 1. Prove the base step let n=1 2^1=2^(1+1)1 False. Someone else suggested that … 
math
Use mathematical induction to show that 3n + 7n − 2 is divisible by 8 for all n 1. [Hint: 7n + 1 is divisible by 2.] 
Algebra ASAP
so this is a fill in on a worksheet and I am having difficulty as the ones I inserted are incorrect can anybody help me how to do it all, sorry it's a long problem. Show that 3^2n − 1 is divisible by 8 for all natural numbers … 
Mathematical Induction
Use mathematical induction to prove that the following is true. 8+11+14...+(3n+5)=1/2n(3n+13), for all n in the set of natural numbers.