As the title says, prove each of the following by induction:
a) for every positive integer
b) for all integers
c) is positive for all
d) divides for all positive integers
Show your work, of course.
Hello RuntyI'll leave all the peripheral bits of the proofs to you. But here are the algebraic proofs that .
a) is the propositional function:
Thenb) is the propositional function:
Thenc) is the propositional function:for
Now
since whenHenced) I'll leave this to you. It's rather similar to (c).
Expand and simplify . Then express the resulting expression in the form , and you'll find that is divisible by .
Can you fill in all the gaps?
Grandad