Question

What is the $12^{th}$ term of the sequence $4,12,36...$ and what is the sum of all $12$ terms?

Answer 1

In this question, we have to find out the required value from the given particulars.
We need to first find out the common ratio and the first term. By dividing the second term by the first term we will get the common ratio. Then putting all the values and the number of terms in the formula of the nth term formula and sum of n terms of the sequence, we can find out the required solution.
Property of G.P.:
The nth term of a G.P sequence is $a{r^{n - 1}}$ .
The sum of the G.P. series is represented by $a + ar + a{r^2} + a{r^3} + ...... + a{r^{n - 1}}$
(Each term is $a{r^k}$ , where$k$ starts at $0$ and goes up to $n - 1$ ) is defined as
$\sum\limits_{k = 0}^{n - 1} {a{r^k} = a\left( {\dfrac{{1 - {r^n}}}{{1 - r}}} \right)}$
Where $a$ is the first term, and $r$ is the factor between the terms (called the "common ratio") and $n$ is the number of terms in the G.P.

Complete step by step solution:
It is given that the sequence is $4,12,36...$ .
We need to find the $12^{th}$ term of the sequence and the sum of all $12$ terms of the sequence $4,12,36...$ .
$a =$ The first term of the arithmetic sequence = $4$ .
$r =$ The common ratio = second term / first term = $\dfrac{{12}}{4} = 3$ .
Since the common ratio is the same once we divide the third term by the second term, so we can apply the formula of G.P. for the given sequence.
$n =$ Number of terms = $12$ .
The $12^{th}$ term of the sequence = $4 \times {3^{12 - 1}} = 7,08,588$
The sum of $12$ terms of the sequence is
${S_{12}} = 4\left( {\dfrac{{1 - {3^{12}}}}{{1 - 3}}} \right)$
Or, ${S_{12}} = 4\left( {\dfrac{{1 - 531441}}{{ - 2}}} \right)$
Or, ${S_{12}} = 4\left( {\dfrac{{ - 531440}}{{ - 2}}} \right)$
Or, ${S_{12}} = 4 \times 265720$
Or, ${S_{12}} = 10,62,880$
Hence, the $12^{th}$ term of the sequence is $7,08,588$ and the sum of $12$ terms of the sequence $4,12,36...$ is $10,62,880$

Note: A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called a common ratio.
In General we write a Geometric Sequence like this: \left\\{ {a,ar,a{r^2},a{r^3},.......} \right\\} ,Where $a$ is the first term, and $r$ is the factor between the terms (called the "common ratio").