Adding exponents with the same base

Question

Can someone explain how they simplified the left hand side to $2^7 - 2$?

$$2^{6} + 2^{5} + 2^{4} + 2^{3} + 2^{2} + 2^{1} =2^{7}−2 = 126$$

Answer 1

Hint: Write down the sum and multiply it by $x$ to obtain two equations (For your question $x=2$).

$$S = x + x^2 + \ldots + x^n$$ $$xS = x^2 + x^3+ \ldots + x^{n+1}$$

Subtract both equations and solve for $S$: $$\implies (1-x)S=x-x^{n+1} \implies S = \dfrac{x-x^{n+1}}{1-x}$$

Hence,

$$S= x+x^2+\ldots + x^n = \dfrac{x-x^{n+1}}{1-x}.$$

Answer 2

Use $$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+...+ab^{n-2}+b^{n-1})$$ for $n=7$, $a=2$ and $b=1$.

Answer 3

Using the binary representation,

\begin{align} 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 =&01111110_b . \end{align}

\begin{align} &01111110_b \\ +&00000010_b\quad=1\cdot2 \\ =& 10000000_b\quad=1\cdot2^7 \end{align}