In: Advanced Math
prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n + 3⋅9^n.
When, n = 0, we have, 2•4ⁿ + 3•9ⁿ = 2 + 3 = 5, divisible by 5
We have, 4
- 1 (mod 5)
So, 4ⁿ
(-1)ⁿ (mod 5) , for all positive integer n
So, 2•4ⁿ
2•(-1)ⁿ (mod 5)
&, 9
- 1 (mod 5)
So, 9ⁿ
(-1)ⁿ (mod 5) , for all positive integer n
So, 3•9ⁿ
3•(-1)ⁿ (mod 5)
So, 2•4ⁿ + 3•9ⁿ
2•(-1)ⁿ + 3•(-1)ⁿ (mod 5)
If, n is even, then, (-1)ⁿ = 1
Then, 2•4ⁿ + 3•9ⁿ
2 + 3
5
0 (mod 5)
So, 5 divides (2•4ⁿ + 3•9ⁿ) when n is even
If, n is odd, then, (-1)ⁿ = -1
Then, 2•4ⁿ + 3•9ⁿ
- 2 - 5
- 5
0 (mod 5)
So, 5 divides (2•4ⁿ + 3•9ⁿ) when n is odd
So, 5 divides (2•4ⁿ + 3•9ⁿ) for all non-negative integer n.