NCERT Class 11-Math՚s: Chapter – 4 Principle of Mathematical Induction Part 5 (For CBSE, ICSE, IAS, NET, NRA 2023)
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Question 15:
State whether the following proof (by mathematical induction) is true or false for the statement.
Proof: By the Principle of Mathematical induction, is true for ,
Again for some . Now we prove that
Answer: False
Since in the inductive step both the inductive hypothesis and what is to be proved are wrong.
4.3 EXERCISE
Short Answer Type
Question 1:
Give an example of a statement which is true for all but and are not true. Justify your answer.
Answer:
Question 2:
Give an example of a statement which is true for all . Justify your answer. Prove each of the statements in Exercises by the Principle of Mathematical Induction:
Answer:
Question 3:
is divisible by , for each natural number .
Answer:
is divisible by
Question 4:
is divisible by 7, for all natural numbers n.
Answer:
Let is divisible by 7;
Question 5:
is divisible by , for all natural numbers .
Answer:
Let is divisible by .
Question 6:
is divisible by 8, for all natural numbers n.
Answer:
is divisible by .
Question 7:
For any natural number n, is divisible .
Answer:
is divisible by .
Question 8:
For any natural number n, is divisible by , where x and y are any integers with .
Answer:
is divisible by
Question 9:
is divisible by , for each natural number .
Answer:
is divisible by ; .
Question 10:
is divisible by 6, for each natural number n.
Answer:
is divisible by
Question 11:
for all natural numbers .
Answer:
is true
Question 12:
for all natural number n.
Answer:
is true
Question 13:
, for all natural numbers .
Answer:
is true
Question 14:
for all natural numbers .
Answer:
is true;
Question 15:
for all natural numbers n.
Answer:
is true;