Question

What is the ${5^{th}}$ term of this sequence? $5,10,20....$
A) $30$
B) $40$
C) $50$
D) $80$

Answer 1

If we look at this sequence we can see that it increases with a constant value means that it is a Geometric sequence. In this problem, we have to find the fifth term of this sequence. First of all, we have to find the common ratio of this sequence. We know that the general form of G.P. and using the formula we have to find the fifth term of the given sequence.
Formula used:
Common ratio $= \dfrac{{{2^{nd}}term}}{{{1^{st}}term}} = \dfrac{{{3^{rd}}term}}{{{2^{nd}}term}} = ....$
The general form of G.P is $a,ar,a{r^2}....$
The ${n^{th}}$term in the G.P is $= t\left( n \right) = a{r^{\left( {n - 1} \right)}}$

Complete step-by-step solution:
The given geometric sequence is $5,10,20...$
First, we have to find the common ratio in the sequence.
Already we know the formula for finding the common ratio.
The common ratio is equal to dividing the given second term by the first term.
Here, common ratio=$= \dfrac{{10}}{5} = \dfrac{{20}}{{10}}$
Simplify the values we get,
Common ratio$= 2$
Now using the general form for G.P we are going to find the fifth term of the sequence.
The general form of G.P is $a,ar,a{r^2},a{r^3},a{r^4}$
Here the fifth term is $a{r^4}$
We know that the values,
$a = 5$; First term
$r = 2$; Common ratio
Now substitute the values, we have
$a{r^4} = 5 \times {2^4}$
Now, expand this we get,
$= 5 \times 2 \times 2 \times 2 \times 2$
$= 10 \times 2 \times 2 \times 2$
$= 10 \times 8$
Now we get the value of the fifth term is $= 80$
At the other method, using the formula of ${n^{th}}$ the term of G.P we can find the value of the fifth term.
Substitute the values in the formula,$t\left( n \right) = a{r^{\left( {n - 1} \right)}}$
$= 5 \times {2^{5 - 1}}$
$= 5 \times {2^4}$
$= 80$
Hence the fifth term of the sequence is $80$.
Hence the answer is option D.

Note: In the geometric sequence, it increases with a constant value and has a common ratio. Correctly using the formula is the simplest way to solve the problem. Finding the common ratio is the main procedure of this problem.