Question

Consider a sequence of numbers given by the definition ${{c}_{1}}=2$, ${{c}_{i}}={{c}_{i-1}}\centerdot 3$, how do you write out the first $4$ terms and how do you find the value of ${{c}_{4}}-{{c}_{2}}$?

Answer 1

For this problem we need to calculate the terms in the given sequence. In the problem we have the value of ${{c}_{1}}=2$. From this value we will calculate ${{c}_{2}}$ by substituting $i=2$ in the given equation ${{c}_{i}}={{c}_{i-1}}\centerdot 3$ and simplifying this. After getting the value of ${{c}_{2}}$, we will calculate the value of ${{c}_{3}}$ by using the value of ${{c}_{2}}$ by following same procedure. Again, we will use the given relation and value of ${{c}_{3}}$ to find the value of ${{c}_{4}}$. After getting all the values we can simply calculate the asked value which is ${{c}_{4}}-{{c}_{2}}$.

Complete step-by-step answer:
Given that, ${{c}_{1}}=2$, ${{c}_{i}}={{c}_{i-1}}\centerdot 3$.
Substituting $i=2$ in the above relation, then we will get
$\begin{aligned} & \Rightarrow {{c}_{2}}={{c}_{2-1}}\times 3 \\\ & \Rightarrow {{c}_{2}}={{c}_{1}}\times 3 \\\ \end{aligned}$
Substituting the value of ${{c}_{1}}=2$ in the above equation, then we will get
$\begin{aligned} & \Rightarrow {{c}_{2}}=2\times 3 \\\ & \Rightarrow {{c}_{2}}=6......\left( \text{i} \right) \\\ \end{aligned}$
Now substituting $i=3$ in the given relation, then we will get
$\begin{aligned} & \Rightarrow {{c}_{3}}={{c}_{3-1}}\times 3 \\\ & \Rightarrow {{c}_{3}}={{c}_{2}}\times 3 \\\ \end{aligned}$
Substituting the value of ${{c}_{2}}=6$ in the above equation, then we will get
$\begin{aligned} & \Rightarrow {{c}_{3}}=6\times 3 \\\ & \Rightarrow {{c}_{3}}=18......\left( \text{ii} \right) \\\ \end{aligned}$
Now substituting $i=4$ in the given relation, then we will get
$\begin{aligned} & \Rightarrow {{c}_{4}}={{c}_{4-1}}\times 3 \\\ & \Rightarrow {{c}_{4}}={{c}_{3}}\times 3 \\\ \end{aligned}$
Substituting the value of ${{c}_{3}}=18$ in the above equation, then we will get
$\begin{aligned} & \Rightarrow {{c}_{4}}=18\times 3 \\\ & \Rightarrow {{c}_{4}}=54......\left( \text{iii} \right) \\\ \end{aligned}$
From the equations $\left( \text{i} \right)$, $\left( \text{ii} \right)$, $\left( \text{iii} \right)$ we can write the first four terms in the sequence as $2$, $6$, $18$, $54$.
Now we can write the value of ${{c}_{4}}-{{c}_{2}}$ as
$\begin{aligned} & \Rightarrow {{c}_{4}}-{{c}_{2}}=54-6 \\\ & \Rightarrow {{c}_{4}}-{{c}_{2}}=48 \\\ \end{aligned}$

Note: For this problem they have asked to calculate the first four terms of the given sequence so we have calculated the values of ${{c}_{2}}$, ${{c}_{3}}$, ${{c}_{4}}$ as we have the value ${{c}_{1}}$ in the problem. If they have asked to calculate the next four terms of the sequence then we need to calculate the value of ${{c}_{5}}$ also.