Answer

Verified

381.3k+ views

**Hint:**Natural numbers are those used for counting and ordering. In common mathematical terminology words which are used for counting are called “cardinal numbers” and used for ordering is called “ordinal number”. It doesn’t include zero (0). Natural numbers are positive integers or non-negative integers.

**Complete step by step solution:**

Given equation,

${(1 + x)^n} \geqslant (1 + nx).$

$P(n):{(1 + x)^n} \geqslant (1 + nx)\,when\,n \in N\,and\,x > - n$

Let n=1

Then

$P(1):(1 + x) \geqslant (1 + x)\,$

which is true

$\therefore P(1)\,$is true.

Let us consider $P(m)\,$is true

$ \Rightarrow {(1 + x)^m} \geqslant (1 + mx)$ …(1)

$\therefore x > - n$

Then

$ \Rightarrow x + n > 0$

Now if we take n=1.

$ \Rightarrow x + 1 > 0$

Multiplying both side by (x+1) in equation 1

$ \Rightarrow {(1 + x)^{m + 1}} \geqslant (1 + mx)(1 + x)$

$ \Rightarrow {(1 + x)^{m + 1}} \geqslant 1 + (m + 1)x + m{x^2}$

$ \Rightarrow {(1 + x)^{m + 1}} \geqslant 1 + {(m + 1)_x}$ …(2)

$\therefore m{x^2} \geqslant 0$

$\therefore P(n)$ is true for all values of $n \in N$ and \[x >-1\].

**Therefore, the value of $n=1$.**

**Note:**

The number system or numeral system is the system of naming. There are various types of number systems. Natural numbers are counting numbers excluding 0. Whole numbers are all natural numbers including 0. Integers include all positive and negative numbers. Rational number which can be expressed in the form of $\dfrac{p}{q}$ where q≠0. Real numbers are those numbers which can be written in decimals also.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which places in India experience sunrise first and class 9 social science CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE