Question

A sequence is defined by the following: $a_1=6$ and $a_n=-1.2\times a_{n-1}$.

What is the fourth term?

• A. $-12.4416$
• B. $-10.368$
• C. $10.368$
• D. $12.4416$

Answer 1

The correct option is B

$-10.368$

Step-1. Explanation for the correct option:

Express the fourth term $a_4$ in terms of $a_1$ using $a_n=-1.2\times a_{n-1}$ repeatedly.

$\begin{array}{rcl}{a}_4& =& -1.2\times a_{4-1}\\ & =& -1.2\times a_3\\ & =& -1.2\times \left(-1.2a_2\right)\\ & =& 1.44\times a_2\\ & =& 1.44\times \left(-1.2a_1\right)\\ & =& -1.728\times a_1\end{array}$

Substitute given value of $a_1=6$ in $a_4=-1.728\times a_1$ and evaluate:

$\begin{array}{rcl}{a}_4& =& -1.728\times 6\\ & =& -10.368\end{array}$

The fourth term is $-10.368$.

Step-2. Explanation for the incorrect options:

Here, Option $\left(A\right)$ is $-12.4416\ne -10.368$, so it is incorrect.

Here Option $\left(C\right)$ is $10.368\ne -10.368$, so it is incorrect.

Option $\left(D\right)$ is $12.4416\ne -10.368$, So it is incorrect.

Hence, Option $\left(B\right)$ $\mathbf{-}\mathbf{10}\mathbf{.}\mathbf{368}$ is the correct answer. Option $\left(A\right)$, Option $\left(C\right)$, and Option $\left(D\right)$ are incorrect.