a (i)

當n=k+1

5^k+1 -1

=5 (5^k) -1

=5(4M+1)-1

=5 (4M) + 4

=4 (5M+1)

(ii)

4^k+1-1

=4(4^k) -1

=4(3M+1)-1

=4(3M)+3

=3(4M+1)

b.

3(5^k+1)+4^k+2-7

=15 (5^k)+16(4^k)-7

=15(4M+1)+16(3M+1)-7             ) <---a部已證 5^k =4M+1 & 4^k =3M+1

=60M+15+48M+16-7

=108M+24

=12(9M+2)