Question

Find the sum of the geometric sequence. 1, 1/4, 1/16, 1/64, 1/256

Answer 1

in a geometric sequence, we get the next term by multiplying the current term by the "common ratio".

now, if we simply divide the any of the terms by the one before it, their quotient is, yeap, you guessed it, is the "common ratio".

anyhow, divide say (1/16) ÷ (1/4) or (1/256) ÷ (1/64), and you'll find it's 1/4.

bearing in mind that there are 5 terms, and the first one is 1.

$\bf \qquad \qquad \textit{sum of a finite geometric sequence} \\\\ S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ a_1=1\\ r=\frac{1}{4}\\ n=5 \end{cases}$

$\bf S_5=1\left( \cfrac{1-\left( \frac{1}{4} \right)^5}{1-\frac{1}{4}} \right)\implies S_5=1\left(\cfrac{1-\frac{1^5}{4^5}}{\frac{3}{4}} \right)\implies S_5=1\left(\cfrac{1-\frac{1}{1024}}{\frac{3}{4}} \right) \\\\\\ S_5=1\left(\cfrac{\frac{1023}{1024}}{\frac{3}{4}} \right)\implies S_5=\left( \cfrac{1023}{1024}\cdot \cfrac{4}{3}\right)\implies S_5=\cfrac{341}{256}$

Answer 2

hope this helped i tried