Question

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

Answer 1

Hint: 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.