I have the following statement: 5|7^k-2^k for all k in N
I want to prove it by induction, so I proceed as follows..
let p(k) be the statement that 7^k-2^k = 5m for m in N and k in P.
Then p(1) is the statement 7^1-2^1=5m for some m in N. This is clearly true for m = 1.
Suppose that p(k) is true for some k in P, That is, 7^k-2^k = 5m for some m in N, then, 7^(k+1) - 2^(k+1) = 5*1^(k+1) for k + 1 in N.
Therefore, p(k+1) is true. Hence p(k) is true for all k in P.
The basis step is straight forward, but I am not sure whether I have done the induction step correctly particularly with the result 5^(k+1). Any criticism welcome. Thanks
5 divides 7^k-2^k
5 divides 7^(k+1)-2^(k+1)
Try to show that P(k+1) will be true if P(k) is true.
7^(k+1)-2^(k+1) = (7)7^k-(2)2^k
Now, if P(k) really is true, what can we say about P(k+1) ?