Test: Introduction To Mathematical Induction - JEE MCQ

# Test: Introduction To Mathematical Induction - JEE MCQ

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Test: Introduction To Mathematical Induction - Question 1

### If P(n) : (2n + 7) < (n + 3)2 then P(3) is

Detailed Solution for Test: Introduction To Mathematical Induction - Question 1

P(n) : (2n + 7) < (n + 3)2
P(3) = [(2(3) + 7) < (3+ 3)2

Test: Introduction To Mathematical Induction - Question 2

### If P(n) is the statement; 4 + 8 + 12 +……+ 4n = 2n (n + 1) then P(6) will be

Detailed Solution for Test: Introduction To Mathematical Induction - Question 2

P(n) = 4 + 8 + 12 +……+ 4n = 2n (n + 1)
P(6) = 4 + 8 + 12 +……+ 4(6)
= 2(6)[(6) + 1)]

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Test: Introduction To Mathematical Induction - Question 3

### If 42x = 4-6 then what is the value of x ?

Detailed Solution for Test: Introduction To Mathematical Induction - Question 3

42x = 4-6

2x = -6

So,  x = -3

Test: Introduction To Mathematical Induction - Question 4

Assuming the truth of P(k) and proving P(k + 1) to be true, for some integer k is known as the _______ .

Detailed Solution for Test: Introduction To Mathematical Induction - Question 4

Assume the inductive hypothesis that we can reach rung k. Then, we can reach rung k + 1.
Hence, P(k) → P(k + 1) is true for all positive integers k

Test: Introduction To Mathematical Induction - Question 5

Every even power of an odd number greater than 1 when divided by 8 leaves 1 as the remainder.The inductive step for the above statement is

Detailed Solution for Test: Introduction To Mathematical Induction - Question 5

Test: Introduction To Mathematical Induction - Question 6

A set S in which x S implies x+1 S is known as a _______ .

Test: Introduction To Mathematical Induction - Question 7

If one of the factor of x2 + x – 20 is (x + 5). Find the other

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Test: Introduction To Mathematical Induction - Question 8

If p(n) is the statement :  12 + 32 + 52 + __ __ + (2n - 1)2    then which one of the following is incorrect?

Detailed Solution for Test: Introduction To Mathematical Induction - Question 8

Correct Answer :- b

Explanation : Let P(n) be the given statement, i.e., P(n) : (2n + 1) < 2n

for all natural

numbers, n ≥ 3. We observe that P(3) is true, since

2.3 + 1 = 7 < 8 = 23

Assume that P(n) is true for some natural number k, i.e., 2k + 1 < 2k

To prove P(k + 1) is true, we have to show that 2(k + 1) + 1 < 2k+1. Now, we have

2(k + 1) + 1 = 2 k + 3

= 2k + 1 + 2 < 2k + 2 < 2k

. 2 = 2k + 1

Thus P(k + 1) is true, whenever P(k) is true.

Hence, by the Principle of Mathematical Induction P(n) is true for all natural numbers, n ≥ 3

Test: Introduction To Mathematical Induction - Question 9

If a statement is to be proved by mathematical induction, then the different steps necessary to prove it are

Detailed Solution for Test: Introduction To Mathematical Induction - Question 9

To prove a statement by induction, we must prove parts 1) and 2) above. The hypothesis of Step 1) -- "The statement is true for n = k" -- is called the induction assumption, or the induction hypothesis. It is what we assume when we prove a theorem by induction.

Test: Introduction To Mathematical Induction - Question 10

In the principle of mathematical induction, which of the following steps is mandatory?

Detailed Solution for Test: Introduction To Mathematical Induction - Question 10

The hypothesis of Step is a must for mathematical induction that is the statement is true for n = k, where n and k are any natural numbers, which is also called induction assumption or induction hypothesis.

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