This was step 1.
Moreover step 1 PROVED that there is AT LEAST one number for which the proposition is true, and there may possibly be more than one. So there is no need to assume anything in step 2.
Incorrect. Step 1 is independent from step 2. One must assume that the statement is true for n = k. And using that, one must show the statement is true for n = k + 1, which would be step 3.
[After step 1 has been shown, the following is presented.]
Assume the statement is true for n = k:
\(\displaystyle 3^{4k} \ - \ 1 \ = \ \) a multiple of 8
\(\displaystyle 3^{4k} \ - \ 1 \ = \ 8M , \ \ \ \)for some integer M.
Show the statement is true for n = k + 1:
\(\displaystyle 3^{4(k + 1)} \ - \ 1 \ = \ \) a multiple of 8
\(\displaystyle 3^{4k + 4} \ - \ 1 \ = \ \) a multiple of 8
Working with the assumption:
\(\displaystyle 3^{4k} \ - \ 1 \ = \ 8M \)
\(\displaystyle 3^4(3^{4k} \ - \ 1) \ = \ 3^4(8M) \)
\(\displaystyle 3^{4k + 4} \ - \ 81 \ = \ 81*8M \)
\(\displaystyle 3^{4k + 4} \ - \ 81 \ + \ 80 \ = \ 81*8M \ + \ 80 \)
\(\displaystyle 3^{4k + 4} \ - \ 1 \ = \ 81*8M \ + \ 80 \)
\(\displaystyle 3^{4k + 4} \ - \ 1 \ = \ 8(81M \ + \ 10) \)
\(\displaystyle 3^{4k + 4} \ - \ 1 \ = \ \) a multiple of 8
Step 4:
Thus, by the Principle of Mathematical Induction, the statement which was to be proved is true.
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