Question

How do you identify the ${10^{th}}$ term of the sequence $2,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 8,{\kern 1pt} {\kern 1pt} {\kern 1pt} 32$ ?

Answer 1

Hint: We are given three terms of a sequence. We have to first identify the type of sequence to find the ${10^{th}}$ term. We will check whether the sequence is in Arithmetic Progression (AP) or in Geometric Progression (GP). The GP has a common ratio while an AP has a common difference between the consecutive terms. Once we identify the type of sequence, we can use the formula for ${n^{th}}$ term to find the ${10^{th}}$ term.

Formula used: ${T_n} = a{r^{n - 1}}$

Complete step-by-step solution:
Given sequence is $2,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 8,{\kern 1pt} {\kern 1pt} {\kern 1pt} 32$. We have to first identify the type of sequence, i.e. we have to identify the relationship between the consecutive terms.
On simple observation we see that the ratio between the consecutive terms is $ - 4$, i.e.
$\dfrac{{ - 8}}{2} = \dfrac{{32}}{{ - 8}} = - 4$
Thus the ratio between the consecutive terms is common, i.e. $ - 4$.
Such a sequence is known as Geometric Progression (GP). A geometric progression is a sequence that has a common ratio between the consecutive terms, i.e. the terms are in the form $a,\,\,\,ar,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a{r^2},{\kern 1pt} {\kern 1pt} {\kern 1pt} a{r^3},...$ , where $a$ is the first term and $r$ is the common ratio.
The ${n^{th}}$ term of a GP is given by the formula,
${T_n} = a{r^{n - 1}}$
The common ratio is $r = - 4$.
The first term of the sequence is $a = 2$.
For ${10^{th}}$ term, $n = 10$.
Putting all the values in the above formula we get,
$
\Rightarrow {T_n} = a{r^{n - 1}} \\
 \Rightarrow 2 \times {( - 4)^{10 - 1}} \\
\Rightarrow 2 \times {( - 4)^9} \\
\Rightarrow 2 \times ( - 262144) \\
 \Rightarrow - 524288 \\
 $
Thus, we get the ${10^{th}}$ term of the given sequence as $ - 524288$.

Note: To find the ${n^{th}}$ term of a sequence we have to first identify the type of the sequence, i.e. whether the sequence is in AP or GP. We can find the ${n^{th}}$ term of a GP using the formula ${T_n} = a{r^{n - 1}}$ for any value of $n \geqslant 1$.