# Inductive Abstract Algebra proofs

• February 4th 2010, 07:13 PM
Zennie
Inductive Abstract Algebra proofs
Prove the $1/(1-x) = 1 + x +x^2 + ... + x^{(n-1)} + x^n/(1-x)$ for all n greater than or equal to 1.

Assuming P(n) is the statement above, P(1) is true.
Induction says that P(n) is true if
A. P(n0) is true, in this case P(1)
B. for all k greater than or equal to n0, if P(k) is true then P(k+1) is true.

However, I can't find a way to show that by assuming P(k) to be true, P(k+1) is therefore also true.
• February 4th 2010, 07:52 PM
Black
$1 + x +\cdots+ x^k+ \frac{x^{k+1}}{1-x}=1 + x +\cdots+ \frac{(1-x)x^k}{1-x} + \frac{x^{k+1}}{1-x}=1 + x +\cdots+ \frac{x^k}{1-x}=\frac{1}{1-x}$