2^n-1 gives the set of all odd natural numbers for all n ∈N. Comment on the given statement. (1)

step 1 Hi, we have given here not 2 to the part n negative 1. The set of all or natural numbers here we

2^n-1 gives the set of all odd natural numbers for all n ∈N....

$2^{n}-1$ gives the set of all odd natural numbers for all $\mathrm{n} \in \mathrm{N}$. Comment on the given statement.
(1) True for all values of $\mathrm{n}$
(2) False
(3) True for only odd values of $\mathrm{n}$
(4) True for only prime values of $\mathrm{n}$

Key Concepts

Function Mapping and Sequence Generation

The concept of function mapping determines how every element of one set (the domain) corresponds to elements of another set (the range). Examining the surjectivity of such mappings, or whether every element of the target set is represented in the output, is a fundamental aspect when considering sequences generated by mathematical expressions.

Mersenne Numbers

Mersenne numbers, typically defined as numbers of the form 2^n - 1, play a significant role in number theory, especially in the context of prime number studies. They highlight how certain forms imposed on numbers create interesting subsets, but not every number in a broader set, such as all odd numbers, can necessarily be obtained by this form.

Exponential Functions

Exponential functions are mathematical functions in which the variable appears in the exponent. Understanding these functions is essential because they exhibit rapid growth and are used to generate sequences, making them a cornerstone in the study of discrete mathematics and number theory.

Domain and Range Analysis

Analyzing the domain and range of a function involves determining the set of possible inputs and the corresponding set of outputs. This concept is crucial when discussing whether a function, such as one defined by an exponential expression, can generate every element of a particular set, like all odd numbers.

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