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I am at the step where I am proving $P(k+1)$:

$$2^k-1+2^k=2^{k+1}-1$$

How am I going to make these equal?

Ps: Just realized this is just an exponent rule, I need coffee.

King Squirrel
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2 Answers2

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Notice: $$\sf \color{red}{2^k}+\color{red}{2^k}-1=2(2^k)-1=2^1(2^k)-1=2^{k+1}-1.$$ Where we used the neat property: $$\sf a^m\cdot a^n=a^{m+n}.$$

Hakim
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Notice looking at the left side of the "equality", we have $$ 2^k-1+2^k=2^k+2^k-1=2^k(1+1)-2=2^k\cdot 2 -1=2^{k+1}-1 $$ This gives us $$ 2^{k+1}-1=2^{k+1}-1 $$ Which is trivially true. So we see that the two sides of the equality are indeed the same.

mathematics2x2life
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  • That is in my first line. I did it using the distributive property in reverse. Notice that $2\cdot 2^k$ is a product of $2$ with $k$ 2's. That means it is a product of $k+1$ 2's, which is $2^{k+1}$. – mathematics2x2life May 19 '14 at 21:04