
Prove that for all positive integers n.
Answer
495.3k+ views
Hint: To solve this question, we will use mathematical induction. In that, we will assume P(n) to be the statement that is to be proved and then by using the Principle of Mathematical Induction, we will get the result.
Complete step by step answer:
We are given n as an integer. The process of solving of proving using Mathematical Induction is given by
1. First we will assume P(n) to be a statement that is to be proved.
2. We will show for n = 1, P(n) = P(1) holds true.
3. Then we will assume for k > 1, P(k) is true.
4. Finally, using the assumed hypothesis that P(k) is true, we will prove that P(k + 1) to be true.
This is the process that is to be followed in this question as well.
Let P(n) be the statement showing that
Let n = 1. Consider, P(n) = P(1).
Hence, 2 > 1. Therefore, P(1) is true.
P(1) is a true statement……(i)
Now for the second step, let us assume k > 1. P(k) is true.
P(k) holds true that is correct or holds true……(ii)
Finally, we have to show that P(k + 1) is true. Consider that To show this is true, we will do as follows. By equation (ii), we have, which is true.
Multiplying 2 on both the sides of the above equation, we have,
Now, as k > 1, adding both sides with k, we have,
Hence, we have,
holds true.
Therefore, P(k + 1) holds true…..(iii)
From equations (i), (ii) and (iii), we have, for all n positive integers is a true statement.
Note: The point where we have used is true if and only if 2 > 0. Now, since 2 > 0 then this implies In short if a > 0, then b > c. Therefore, ab > ac. Also, at the step where we used k > 1, k + k > k + 1. This was true as k was also a positive integer. If k was not a positive integer, here then k > 1, k + k > k + 1 would be true as well as false.
Complete step by step answer:
We are given n as an integer. The process of solving of proving using Mathematical Induction is given by
1. First we will assume P(n) to be a statement that is to be proved.
2. We will show for n = 1, P(n) = P(1) holds true.
3. Then we will assume for k > 1, P(k) is true.
4. Finally, using the assumed hypothesis that P(k) is true, we will prove that P(k + 1) to be true.
This is the process that is to be followed in this question as well.
Let P(n) be the statement showing that
Let n = 1. Consider, P(n) = P(1).
Hence, 2 > 1. Therefore, P(1) is true.
P(1) is a true statement……(i)
Now for the second step, let us assume k > 1. P(k) is true.
P(k) holds true that
Finally, we have to show that P(k + 1) is true. Consider that
Multiplying 2 on both the sides of the above equation, we have,
Now, as k > 1, adding both sides with k, we have,
Hence, we have,
Therefore, P(k + 1) holds true…..(iii)
From equations (i), (ii) and (iii), we have,
Note: The point where we have used
Latest Vedantu courses for you
Grade 10 | MAHARASHTRABOARD | SCHOOL | English
Vedantu 10 Maharashtra Pro Lite (2025-26)
School Full course for MAHARASHTRABOARD students
₹33,300 per year
Recently Updated Pages
Master Class 11 Economics: Engaging Questions & Answers for Success

Master Class 11 Accountancy: Engaging Questions & Answers for Success

Master Class 11 English: Engaging Questions & Answers for Success

Master Class 11 Social Science: Engaging Questions & Answers for Success

Master Class 11 Physics: Engaging Questions & Answers for Success

Master Class 11 Biology: Engaging Questions & Answers for Success

Trending doubts
How many moles and how many grams of NaCl are present class 11 chemistry CBSE

How do I get the molar mass of urea class 11 chemistry CBSE

Plants which grow in shade are called A Sciophytes class 11 biology CBSE

A renewable exhaustible natural resource is A Petroleum class 11 biology CBSE

In which of the following gametophytes is not independent class 11 biology CBSE

Find the molecular mass of Sulphuric Acid class 11 chemistry CBSE
