Question

How do you write a rule for the ${{n}^{th}}$ term of the geometric sequence given the two terms ${{a}_{3}}=5$ and ${{a}_{6}}=5000$ ?

Answer 1

We start solving the problem by writing the general expression for the ${{n}^{th}}$ term of a geometric sequence is $a{{r}^{n-1}}$ where a is the first term of the sequence and r is the constant common ratio. We first put $n=3$ and equate it to $5$ and then put $n=6$ and equate it to $5000$ . Finally, we divide the two equations to get r and after that a. These two values are used to rewrite general expressions for the ${{n}^{th}}$ term.

Complete step by step answer:
Sequence is an enumerated collection of objects in which repetitions are allowed and where the order of objects is the most important. By objects, we mean generally numbers. Sequence may follow a specific pattern or can even be completely random. If the sequence follows a certain pattern, then the sequence can be of various types like arithmetic sequence, geometric sequence and so on.
The general expression for the ${{n}^{th}}$ term of a geometric sequence is $a{{r}^{n-1}}$ where a is the first term of the sequence and r is the constant common ratio. In the given problem, we are given the third and the sixth terms of the sequence. So, using the general expression for the ${{n}^{th}}$ term, we get,
$\begin{aligned} & \Rightarrow {{a}_{3}}=a{{r}^{3-1}}=5 \\\ & \Rightarrow a{{r}^{2}}=5....\left( 1 \right) \\\ \end{aligned}$

& \Rightarrow {{a}_{6}}=a{{r}^{6-1}}=5000 \\\ & \Rightarrow a{{r}^{5}}=5000....\left( 2 \right) \\\ \end{aligned}$$ Dividing equation $\left( 2 \right)$ by equation $\left( 1 \right)$ , we get, $\Rightarrow \dfrac{a{{r}^{5}}}{a{{r}^{2}}}=\dfrac{5000}{5}$ Simplifying the above equation by cancelling the common terms in the numerator and the denominator, the above expression thus becomes, $\Rightarrow {{r}^{3}}=1000$ Taking cube roots on both sides of the above equation, we get, $\Rightarrow r=10$ Putting this value of r in equation $\left( 1 \right)$ , we get, $\begin{aligned} & \Rightarrow a{{\left( 10 \right)}^{2}}=5 \\\ & \Rightarrow a=0.05 \\\ \end{aligned}$ **Thus, we can conclude that the general rule for the ${{n}^{th}}$ term will be $\left( 0.05 \right){{\left( 10 \right)}^{n-1}}$** **Note:** In order to solve these problems, we must have some basic knowledge of the various types of sequences. So, we must be thorough with the sequences. We must be careful with the general expression of the ${{n}^{th}}$ term which is $a{{r}^{n-1}}$ . Students often misinterpret it as $a{{r}^{n}}$ which leads to wrong answers.