Question

How do you find the recursive formula that describes the sequence: $3,7,15,31,63...$?

Answer 1

Hint:The given question requires us to find the general formula for the terms of the sequence given to us. Hence, we have to generalize a formula for the terms of the given sequence. We have to find out whether the given series or sequence is an arithmetic progression, a geometric progression, a harmonic progression, an arithmetic geometric progression or a special type of series.

Complete step by step answer:
The given problem puts our analytical skills to test. We have to first identify the nature of the given sequence or series and then find a generalized formula for the terms of the sequence.
The sequence given to us is: $3,7,15,31,63...$.
First checking the given series for arithmetic progression. The difference between the first two terms of the sequence is $4$ and the difference between the second and the third term of the sequence is $8$.
Since the difference between the consecutive terms of the series is not equal, hence it is not an arithmetic progression.
Now, checking the series for Geometric progression. The ratio of first two terms is $\left( {\dfrac{7}{3}} \right)$ and the ratio of third term to the second term is $\left( {\dfrac{{15}}{7}} \right)$. Hence, it is not a geometric progression.
But looking at the difference between any two consecutive terms, we can say that the difference between two consecutive terms is in geometric progression. So, it is an arithmetic geometric progression.
If we observe carefully, we see that each term is $1$ less than a power of $2$.
Hence, the general term of the sequence is: $\left( {{2^{n + 1}} - 1} \right)$.

Note: In such a type of question, we should first find out the nature of the series and then try to figure out the general term of the series. In this way, we would have an idea beforehand of what the formula for the general term of the sequence would look like.