Question

What is the $nth$ term of $1,7,17,31 \ldots$?

Answer 1

Here we have to find the $nth$ of the sequence $1,7,17,31 \ldots$. Sequence is defined as the compiled list of objects or events that are placed in such a way that each member of a given sequence comes before or after every other member in the list of sequences.

Complete step by step answer:
Here in the given problem we have to find the $nth$ term of the sequence $1,7,17,31 \ldots$
Firstly, we need to find a pattern between the value and the place in the sequence that value holds.
We have
$1 \to 2$
$2 \to 7$
$3 \to 17$
$4 \to 31$
So, in the given sequence the numbers are one less than twice a perfect square
$1 \to 1 = 2 \times {(1)^2} - 1$
$2 \to 7 = 2 \times {(2)^2} - 1$
$3 \to 17 = 2 \times {(3)^2} - 1$
$4 \to 31 = 2 \times {(4)^2} - 1$
Therefore, the $nth$ of sequence $1,7,17,31 \ldots$ can be calculated by the formula $n \to 2{n^2} - 1$.

Note:
A sequence is a particular format of elements in some definite order, whereas a series is the sum of the elements of the sequence. In a Sequence, the order of the elements are definite, but in series the order of elements is not fixed. A sequence is represented as $5,6,7,8, \ldots n$ whereas a series is represented as $5 + 6 + 7 + 8 + 9 + \ldots n$.