Question

How do I find the ${n^{th}}$ term rule for $1,3,9,27...$ ?.

Answer 1

In this we need to find the ${n^{th}}$ term rule for $1,3,9,27...$ . Here, we will find the difference between each term and compare with the consecutive terms of the sequence. By which we will find the pattern of the sequence. Hence, we will determine the ${n^{th}}$ term of the sequence which will be the required ${n^{th}}$ term rule.

Complete step-by-step solution:
Here, we need to find the ${n^{th}}$ term rule for $1,3,9,27...$ .
Now, look at the series, we can see that the terms in the sequence are $3$ times the number right before it.
So, the first term in the series can be written as ${3^0} = 1$ .
The second term can be written as ${3^1} = 3$ .
The third term can be written as ${3^2} = 9$ .
Similarly, the fourth term can be written as ${3^3} = 27$ .
Similarly we can find the value of consecutive terms of the sequence.

Now, the sequence can be written as ${3^0},{3^1},{3^2},{3^3},...$ .

Therefore, we can see here that the base is common in the sequence i.e., $3$ and the exponents of the each term are increasing by $1$ i.e., the exponent of the first term is $0$ , the exponent of the second term is $1$ , the exponent of the third term is $2$ , the exponent of the fourth term is $3$ and so on. From this we can see that the value of exponents is one less than the number of terms. Thus, we can say from this the exponent of the ${n^{th}}$ term will be $\left( {n - 1} \right)$ .

Therefore, the ${n^{th}}$ of the sequence can be written as ${3^{\left( {n - 1} \right)}}$ , which is the required ${n^{th}}$ term rule for the sequence $1,3,9,27...$ .

Note In these types of questions, first we need to determine the pattern of the sequence given. By which we can easily determine the ${n^{th}}$ term rule. However, we can use another method for determining the sequence, i.e., the ratios between the consecutive terms is equal to $3$ . This is a characteristic of a geometric series.