Question

Check if the sequence is an AP $1, 3, 9, 27, ......$
A) Yes, it is an AP
B) No
C) Data Insufficient
D) Ambiguous

Answer 1

Here, we have to find whether the given sequence is an A.P. We will first find the common difference between the consecutive terms in the given series. By using the common difference, we will find the sequence is an A.P or not.

Formula Used:
Common Difference is given by the formula $d = {t_{n + 1}} - {t_n}$ where ${t_{n + 1}}$ is the $\left( {n + 1} \right)th$ term and ${t_n}$ is the $n^{\text{th}}$ term.

Complete step by step solution:
We are given with a series $1, 3, 9, 27, ......$
We will find whether the sequence is an A.P.
A series is said to be in AP if the common difference between the consecutive terms are the same.
Therefore, we will first find the common difference between the consecutive terms.
Common Difference is given by the formula $d = {t_{n + 1}} - {t_n}$.
Common Difference between the first term and the second term ${d_1} = 3 - 1 = 2$
Common Difference between the second term and the third term ${d_2} = 9 - 3 = 6$
Common Difference between the third term and the fourth term ${d_3} = 27 - 9 = 18$
Comparing the common difference, we get ${d_1} \ne {d_2} \ne {d_3}$.
Since the Common Difference between the consecutive terms is not the same always, thus the sequence is not an A.P.
Therefore, the series $1, 3, 9, 27,......$ is not an A.P.

Thus, option (B) is the correct answer.

Note:
We know that Arithmetic progression is a sequence of numbers where the difference between the two consecutive numbers is a constant. If the same number is added or subtracted from each term of an A.P., then the resulting terms in the sequence are also in A.P. but with the same common difference. A real life example of AP is when we add a fixed amount in our money bank every week. Similarly, when we ride a taxi, we pay an amount for the initial kilometer and pay a fixed amount for all the further kilometers, this also turns out to be an AP.