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

A geometric sequence is characterized by a common ratio.

The sum of the geometric terms is: $\mathbf{\frac {341}{256}}$

The sequence is given as:

$\mathbf{1, \frac 14, \frac 1{16}, \frac 1{64}, \frac 1{256}}$

First, we calculate the common ratio

This is calculated by dividing the second term by the first term

$\mathbf{r = \frac 14 \div 1}$

$\mathbf{r = \frac 14}$

The sum of terms of a geometric sequence is:

$\mathbf{S_n = \frac{a(1 - r^n)}{1 - r}}$

Where

n = 5 --- the number of terms

a = 1 -- the first term

So, we have:

$\mathbf{S_5 = \frac{1 \times (1 - (\frac 14)^5)}{1 - \frac 14}}$

$\mathbf{S_5 = \frac{1 \times (1 - \frac 1{1024})}{\frac 34}}$

$\mathbf{S_5 = \frac{1 - \frac 1{1024}}{\frac 34}}$

Subtract

$\mathbf{S_5 = \frac{\frac {1023}{1024}}{\frac 34}}$

Express as products

$\mathbf{S_5 = \frac {1023}{1024} \times {\frac 43}}$

$\mathbf{S_5 = \frac {341}{256}}$

Hence, the sum of the geometric terms is: $\mathbf{\frac {341}{256}}$

Read more about geometric sequence at:

https://brainly.com/question/18109692