CLASS_11_MATHS_SOLUTIONS_NCERT

Class XI Chapter 4 – Principle of Mathematical Induction Maths
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Thus, Pk 1
is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all numbers i.e., N.
Question 23:
Prove the following by using the principle of mathematical induction for all
n n
41 14 is a multiple of 27.
Solution 23:
Let the given statement be
:41 14
P n
n
It can be observed that
n
Pn
is a multiple of 27.
Pn
, i.e.,
is true for n = 1
Since
, which is a multiple of 27.
Let P(k) be true for some positive integer k, i.e.,
k
41 14
1 1
41 14 27
k
is a multiple of 27
k k
41 14 27 m,
We shall now prove that
Consider
41 14
k1 k1
k
k
41 .4114 .14
m N...... 1
P k 1
k k k k
41 41 14 14 14 14
k
41.27 m 14 4114
41.27m
27.14 k
27 41m
14 k
27 r,
where
Therefore,
Thus,
P k 1
r
41 14
k1 k1
41m14 k
is true whenever P(k) is true.
is a natural number.
is a multiple of 27.
is true whenever P(k) is true.
n
N:
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers
i.e., N.