Show that replacing k with k + 1 in 1 - 1/2k gives an expression equivalent to 1 - 1/2k + 1/2k+1
We can also do this with partial fractions and then it will go in the correct direction.
\(1-\frac{1}{2^{k+1}}\\ =1+\;\;\frac{-1}{2*2^k}\\ \mbox{There are some integers A and B such that}\\ =1+\;\;\frac{A}{2^k}+\frac{B}{2^{k+1}}\\ =1+\;\;\frac{2A}{2^{k+1}}+\frac{B}{2^{k+1}}\\ =1+\;\;\frac{2A}{2^{k+1}}+\frac{B}{2^{k+1}}\\ =1+\;\;\frac{2A+B}{2^{k+1}}\\ \qquad SO\\ \qquad 2A+B=-1\\ \qquad \mbox{One solution to this is }\;\;A=-1\;\;and\;\;B=+1\\ =1+\;\;\frac{-1}{2^k}+\frac{1}{2^{k+1}}\\ =1-\;\;\frac{1}{2^k}+\frac{1}{2^{k+1}}\\ \)
[Any A and B such that 2A+B= -1 will also be true ]
