Can anyone please help me with the following question:

Prove by mathematical induction that 6^n + 4 is a multiple of 5, for nEN.

Take n=1 6^1+4 =>10 which is divisible by 5.

6^n+4 is divisible by 5 for all nEN.
Take n= n+1
6^n+1 +4 => 6^n*6 +4 => 6[6^n +4]
as 6^n+4 is divisible by 5 for all nEN
6*[multiples of 5] also divisible by 5
Hence 6^n+4 is divisible by 5 for all nEN.

mmmmuj

Follow the standard 3-step method

1. show it to be true for n=1
6^1 + 4 = 10 which is divisible by 5

2. Assume it to be true for n = k , that is
6^k + 4 is divisible by 5

3. show that it is also true for n = k+1
that is,
6^(k+1) + 4 is divisible by 5
Use the fact that if 2 numbers are both divisible by some number , then their difference is also divisible by that number. e.g. bot 35 and 14 are divisible by 7, so is 35-14 or 21 divisible by 7

(6^(k+1) + 4) - (6^k + 4)
= 6^(k+1) - 6^k
= 6^k(6^1 - 1)
= 5(6^k)
which is a multiple of 5, thus divisible by 5

Therefore 6^(k+1) + 4 is divisible by 5

Sure! Let's prove by mathematical induction that for any natural number n, the expression 6^n + 4 is a multiple of 5.

First, we will establish the base case. Let's substitute n = 1 into the expression and see if it is a multiple of 5:

For n = 1:
6^1 + 4 = 6 + 4 = 10

As we can see, 10 is indeed a multiple of 5 (10 = 2 * 5). So, the base case holds.

Now, let's assume that for some kEN, the expression 6^k + 4 is a multiple of 5. We will use this assumption to prove that the expression holds true for the next case, k+1.

For k+1:
6^(k+1) + 4

Using the property of exponents, we can rewrite this as:
(6^k * 6) + 4

Now, by our assumption, we know that 6^k + 4 is divisible by 5. In other words, there exists an integer m such that: 6^k + 4 = 5m.

Now, substitute this back into our expression:
(5m + 6) + 4 = 5m + 10

We can write 5m + 10 as 5(m + 2). This shows that 5m + 10 is indeed a multiple of 5.

So, we have shown that if the expression is true for k, then it is also true for k+1. Thus, by the principle of mathematical induction, the expression 6^n + 4 is a multiple of 5 for all nEN.