Chapter 1, Problem 5MCQ

Solution

Find the series of $0.4,1.2,3.6,10.8,\mathrm{............},$

The value is $32.4$

The series become $0.4,1.2,3.6,10.8,\underset{¯}{32.4}$

Given:

The geometric series: $0.4,1.2,3.6,10.8,\mathrm{............},$

Concept used:

For any geometric progression, the $n\text{th}$ term is defined is as follows:

  $a_n=a{r}^{n-1}$

Where, $a$ is the first time of the arithmetic progression.

  $r=\frac{a_i}{a_{i-1}}$ is the common ratio of the arithmetic progression

Calculation:

In the given arithmetic progression, $0.4,1.2,3.6,10.8,\mathrm{............},$

  $a=0.4$ is the first time of the arithmetic progression.

  $r=\frac{a_i}{a_{i-1}}$ is the common ratio of two consecutive term of the geometric progression which is calculated as follows:

  $\begin{array}{l}r=\frac{a_i}{a_{i-1}}\\ r=\frac{a_2}{a_1}\left[\text{take,}\text{i=2}\right]\\ r=\frac{1.2}{0.4}\\ r=3\end{array}$

Here the required next term is $5\text{th}$ term of the sequence.

Hence the required term is calculated as follow:

  $\begin{array}{l}{a}_n=a{r}^{n-1}\\ a_5=0.4\times {\left(3\right)}^{5-1}\\ a_5=0.4\times 3\times 3\times 3\times 3\\ a_5=32.4\end{array}$

Hence $5\text{th}$ term is $a_5=32.4$

Conclusion:

The required next term is $32.4$