Equivalence between sum expression and power expression

Question

This is a bit silly, but how can I show that $$ \frac{1}{4}\left(3^{x+1}+5\right) = \frac{1}{2}\left(3+\sum_{i=0}^x 3^i\right) $$

Answer 1

Hint:

By the sum formula for a finite geometric series, we have,

$$\sum_{r=0}^x 3^r=\frac{3^{x+1}-1}{3-1}$$

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Let us prove the formula for ourselves. Take the sum as $S$. Then,

$$3S=\sum_{r=1}^{x+1}3^r\implies 3S-S=3^{x+1}-3^0=3^{x+1}-1\\ \implies S=\frac{3^{x+1}-1}{3-1}$$