MP Board Class 11th Maths Important Questions Chapter 4 Principle of Mathematical Induction
MP Board Class 11th Maths Important Questions Chapter 4 Principle of Mathematical Induction
MP Board Class 11th Maths Important Questions Chapter 4 Principle of Mathematical Induction
Principle of Mathematical Induction Long Answer Type Questions
Question 1.
For n > 1, prove that with the help of principle of mathematical induction :
12 + 22 + 32 + 42 + …………. + n2 = 
(NCERT)
Solution:
∴ p(n) is true for n = k + 1.
Hence, p(n) is true for all values of n ∈ N.
Question 2.
For n ≥ 1 prove that :
Solution:
Let
∴ p(n) is true for n = k + 1.
Hence, p(n) is true for all values of n ∈ N.
Instruction:
All questions from 3 to 7 when n∈N prove the statement by using principle of mathematical induction :
Question 3.
13 + 23 + 33 + …………. + n3 = 
(NCERT)
Solution:
∴ Given function is true for n = m + 1.
Hence, p(n) is true for all values of n ∈ N.
Question 4.
∴ The Given function is true for n = m + 1.
Hence, p(n) is true for all values of n ∈ N.
Question 5.
∴ p(n) is true for n = k + 1.
Hence, p(n) is true for all values of n ∈ N.
Question 6.
∴ p(n) is true for n = k + 1.
Hence, p(n) is true for all values of n ∈ N.
Question 7.
∴ Given function is true for n = 1.
Let the function be true for n = k
∴ The given function is true for n = k + 1. Hence the given function is true for all values of n ∈ N.
Question 8.
Prove that (41)n – (14)n is a multiple of 27 with the help of principle of mathematical induction. (NCERT)
Solution:
Let P(n) = (41)n – (14)n
For n = 1,
P(1) = (41)1 – (14)1= 27
Which is multiple of 27.
Let P(ri) be true for n = k.
Then, (41)k – (14)k = 27m
Put n = k +1 in P(n)
(41)k+1 – (14)k+1 = (41)k.41 – (14)k. 14
= (41)k.41 – 41.(14)k + 41.(14)k – (14)k.14
= 41[(41)k – (14)k]+(14)k [41 – 14]
= 41 x 27m + (14)k x 27 = 27[41 x m + (14)k]
Which is multiple of 27.
If the given function P(n) is true for n = k then P(n) will be true for U n = k +1 also.
Hence the given function is true for all values of n ∈ N.