Question

How do you write a rule for the nth term of the geometric term given the two terms \[{a_3} = 24\], \[{a_5} = 96\]?

Answer 1

Hint: Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. The geometric sequence formula refers to determining the nth term of a geometric sequence. To write a rule for the nth term of the geometric term, we need to consider the given two terms and apply the general formula to get the rule.
Formula used:
  \[{a_n} = {a_1} \cdot {r^{n - 1}}\]
In which,
  \[{a_n}\]is the \[{n^{th}}\]term.
  \[{a_1}\] is the first term.
r is the common ratio.
  \[GM = \sqrt {ab} \]

Complete step by step solution:
 Let us write the given data:
  \[{a_3} = 24\], \[{a_5} = 96\]
We are given \[{a_3}\] and \[{a_5}\], so we can easily find out \[{a_4}\]in order to get the value of r as:
  \[{a_3},{a_4},{a_5}\]
  \[ \Rightarrow 24,{a_4},96\]
To find \[{a_4}\], we need to just calculate the GM (geometric mean) and the formula is given as:
  \[GM = \sqrt {ab} \]
  \[{\left( {{a_4}} \right)^2} = \sqrt {{a_3} \cdot {a_5}} \]
Substitute the given values of \[{a_3}\] and \[{a_5}\]as:
  \[ \Rightarrow {a_4} = \pm \sqrt {24 \cdot 96} \]
  \[ \Rightarrow {a_4} = \pm \sqrt {2304} \]
We know that, \[\sqrt {2304} = 48\], hence we get:
  \[ \Rightarrow {a_4} = \pm 48\]
Hence, the three terms of \[{a_3},{a_4},{a_5}\] are: \[24,48,96\] or \[24, - 48,96\], since we obtained the value of \[{a_4}\] as \[ \pm 48\].
By, this we get the value of r by taking \[{a_4}\] as +48 and -48 i.e.,
  \[ \Rightarrow r = \dfrac{{{a_4}}}{{{a_3}}} = + \dfrac{{48}}{{24}} = + 2\]
And also,
  \[ \Rightarrow r = \dfrac{{{a_4}}}{{{a_3}}} = - \dfrac{{48}}{{24}} = - 2\]
Hence, the value of r is \[r = \pm 2\].
The general formula for a geometric sequence is \[{a_n} = {a_1} \cdot {r^{n - 1}}\], where \[{a_n}\]is the \[{n^{th}}\]term, \[{a_1}\] is the first term, and r is the common ratio.
After we have the common ratio r, now we need to find \[{a_1}\] as:
Given, we have \[{a_3} = 24\], \[{a_5} = 96\], substitute them into the general formula as \[{a_n} = {a_1} \cdot {r^{n - 1}}\]:
  \[{a_3} = {a_1} \cdot {r^{3 - 1}}\]
  \[ \Rightarrow 24 = {a_1} \cdot {r^2}\] ……………….. 1
And,
  \[{a_5} = {a_1} \cdot {r^{5 - 1}}\]
  \[ \Rightarrow 96 = {a_1} \cdot {r^4}\] …………………. 2
Now that we have the value of r, we can find the value of \[{a_1}\] using the equation 1 i.e., \[24 = {a_1} \cdot {r^2}\], we get:
  \[24 = {a_1} \cdot {r^2}\]
  \[ \Rightarrow 24 = {a_1} \cdot {\left( { \pm 2} \right)^2}\]
  \[ \Rightarrow 24 = {a_1} \cdot 4\]
  \[ \Rightarrow {a_1} = \dfrac{{24}}{4} = 6\]
Hence, formula for the sequence is,
  \[{a_n} = 6 \cdot {2^{n - 1}}\] or \[{a_n} = 6 \cdot {\left( { - 2} \right)^{n - 1}}\].
To verify if these are correct, you can write out the first few terms and see if they match the information given data.
  \[{a_n} = 6 \cdot {2^{n - 1}}\]
The common ratio is 2, so start with 6 and multiply each term by 2 as:
  \[ \Rightarrow 6,12,24,48,96\]
Now, \[{a_n} = 6 \cdot {\left( { - 2} \right)^{n - 1}}\]
The common ratio is -2, so start with 6 and multiply each term by -2 as:
  \[ \Rightarrow 6, - 12,24, - 48,96\]
Hence, in both of these formulas we get:
  \[{a_3} = 24\] and \[{a_5} = 96\]
Therefore, a rule for the nth term of the geometric term given the two terms is:
  \[{a_n} = 6 \cdot {2^{n - 1}}\] or \[{a_n} = 6 \cdot {\left( { - 2} \right)^{n - 1}}\].
So, the correct answer is “ \[{a_n} = 6 \cdot {2^{n - 1}}\] or \[{a_n} = 6 \cdot {\left( { - 2} \right)^{n - 1}}\]”.

Note: It is to be noted that when we divide any succeeding term from its preceding term, then we get the value equal to common ratio. If we are asked to find the sum of N term of Geometric Progression then we need to apply the formula as: \[{S_n} = a + ar + a{r^2} + ..... + a{r^{n - 1}}\], in which a is the first term, r is the common ratio and n is the number of terms.