Question

What is the formula for the nth term for the example \[6,12,24,48,96\]?

Answer 1

Hint: Here, the given question is to find the formula for the nth term of the given sequence. To find out which formula is applicable to solve the given sequence, we just need to figure out what progression is suitable, i.e., whether the Arithmetic Progression or Geometric Progression. To know that we need to do the common difference or common ratios for a given sequence. From then, we can know what formula is applied for the nth term to the given sequence.

Complete step-by-step solution:
Before solving this, we must have to know about the basics of arithmetic and geometric progression.
The ‘Arithmetic Progression’ is defined as a sequence of numbers in order in which the difference of any two consecutive numbers is equal and also a constant value and it is abbreviated as AP.
In AP, we must know about the three main terms,
Common difference\[\left( d \right)={{a}_{2}}-{{a}_{1}}\]
Nth term (\[{{a}_{n}}\])
Common differences could be positive, negative or zero.
Nth term,\[{{a}_{n}}=a+\left( n-1 \right)d\]…………………. (1)
The ‘Geometric Progression’ is defined as if each term is a multiple of the next term then this sequence is said to be in geometric progression (G.P).
In GP,
Nth term, \[{{x}_{n}}=x\times {{r}^{n-1}}\] ………………………. (2)
Nth term,\[{{x}_{n}}\]
Common ratio \[\left( r \right)=\dfrac{{{x}_{2}}}{{{x}_{1}}}\]
First term, \[x\]
Let us solve the given question,
Given sequence is \[6,12,24,48,96\]
In this, first term \[{{x}_{1}}=6\]
Second term,\[{{x}_{2}}=12\]
Third term, \[{{x}_{3}}=24\]
To find out which formula should be for nth term, we need to do a common difference and common ratio for the given sequence.
Common difference,
\[{{d}_{1}}={{x}_{2}}-{{x}_{1}}\]
\[\Rightarrow {{d}_{1}}=12-6=6\]
\[{{d}_{2}}={{x}_{3}}-{{x}_{2}}\]
\[\Rightarrow {{d}_{2}}=24-12=12\]
\[\therefore {{d}_{1}}\ne {{d}_{2}}\]
Here, the common difference is not alike, so Arithmetic Progression is not used.
Common ratio,
\[{{r}_{1}}=\dfrac{{{x}_{2}}}{{{x}_{1}}}\]
\[\Rightarrow {{r}_{1}}=\dfrac{12}{6}=2\]
\[{{r}_{2}}=\dfrac{{{x}_{3}}}{{{x}_{2}}}\]
\[\Rightarrow {{r}_{2}}=\dfrac{24}{12}=2\]
\[\therefore {{r}_{1}}={{r}_{2}}\]
Therefore, the common ratios are the same, hence we can choose the Geometric Progression.
In Geometric Progression (GP),
Nth term formula is, \[{{x}_{n}}=x\times {{r}^{n-1}}\]
Here the first term is x= 6 and the common ratio $\left(r\right)=2$.
Hence the general formula for nth term we get as \[{{x}_{n}}=6\times {{2}^{n-1}}\]
The Nth term for Geometric Progression is used to solve the given sequence, \[6,12,24,48,96\].

Note: The sequence is an increasing sequence where the common differences or common ratio is a positive. The common difference will never be calculated according to the difference of greater number from the lesser number.