Question

Which one number will complete the following number sequence ?
$2$,$6$, $14$, $30$, $62$, ?, $254$

Answer 1

A sequence in mathematics is an enumerated collection of objects in which repetitions are permitted and order is important. It has members, just like a set. The length of the sequence is defined by the number of elements.

Complete step by step answer:
Unlike a set, the same elements can appear multiple times in a sequence at different positions, and the order does matter. In mathematical terms, a sequence is a function whose domain is either the set of natural numbers (for infinite sequences) or the set of the first n natural numbers (for finite sequences) (for a sequence of finite length n).
For example, ($M,\text{ }A,\text{ }R,\text{ }Y$) is a letter sequence in which the letter 'M' comes first and the letter 'Y' comes last. This sequence is distinct from the others ($A,\text{ }R,\text{ }M,\text{ }Y$). Also, the sequence ($1$,$1$,$2$,$3$,$5$,$8$) is valid because it contains the number $1$ in two different positions. Sequences can be finite or infinite, such as the sequence of all even positive integers, as shown in these examples ($2$,$4$,$6$, ...).
A geometric progression, also known as a geometric sequence, is a non-zero number sequence in which each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. The sequence $2$, $6$, $18$, $54$,..., for example, is a geometric progression with a common ratio of $3$.
This one is very straightforward. Take a look at the first sequence of differences. $2$,$6$,$14$,$30$,$62$,?,$254$.
$4,8,16,32$is a number that can be found in the numbers $4,8,16,32$. Take on the appearance of G.p.
Check the next term of G.P.=$64$, and it is $128$ after that.
$62\text{ }+\text{ }64$ equals 126, and $126\text{ }+\text{ }128$ equals $254$.
As a result, we've discovered the missing term: $126$. Another way to solve this problem is to realize that the sequence can be created by multiplying the previous term by $2$ and adding $2$, i.e.,
$2\text{ }\times \text{ }2\text{ }+\text{ }2\text{ }=\text{ }6$
$6\text{ }\times \text{ }2\text{ }+\text{ }2\text{ }=\text{ }14$ and so on.
Thus, the answer is $126$.

Note:
Recursion is commonly used to define sequences whose elements are directly related to the previous elements. This differs from the definition of element sequences as functions of their positions.To define a recursive sequence, you'll need a recurrence relation rule to build each element in terms of the ones before it.