Question

Consider the following sequence of numbers.

3,-9,27,-81,...
I need common ratio & sum of 1st five terms

Answer 1

Answer:

Please check the explanation

Step-by-step explanation:

Given the sequence

$3,-9,27,-81,...$

A geometric sequence has a constant ratio and is defined by

$a_n=a_0\cdot r^{n-1}$

Computing the ratios of all the adjacent terms

$r=\frac{a_n+1}{a_n}$

$\frac{-9}{3}=-3,\:\quad \frac{27}{-9}=-3,\:\quad \frac{-81}{27}=-3$

The ratio of all the adjacent terms is the same and equal to

$r=-3$

Therefore, the common ratio is:

• r = -3

Determining the sum of 1st five terms

As the first element is

$a_1=3$

$r=-3$

Geometric sequence sum formula is given by

$a_1\frac{1-r^n}{1-r}$

Plugin the values

$n=5,\:\spacea_1=3,\:\spacer=-3$

$a_1\frac{1-r^n}{1-r}=3\cdot \:\frac{1-\left(-3\right)^5}{1-\left(-3\right)}$

            $=3\cdot \frac{1-\left(-3\right)^5}{1+3}$

             $=\frac{\left(1-\left(-3\right)^5\right)\cdot \:3}{1+3}$

             $=\frac{732}{1+3}$       ∵

             $=\frac{732}{4}$

              $=183$

Therefore, the sum of the first five terms of the sequence is: 183