CLASS_11_MATHS_SOLUTIONS_NCERT

Class XI Chapter 4 – Principle of Mathematical Induction Maths
______________________________________________________________________________
Consider
1.2.3 2.3.4 ..... k k 1 k 2 k 1 k 2 k 3
1.2.3 2.3.4 ..... k k 1k 2 k 1k 2 k
3
1 2 3
k k k k
k 1k 2k 3 Using
i
4
k
k 1 k 2 k 3 1
4
k 1k 2k 3k
4
4
Thus,
k 1 k 11 k 1 2 k 1
3
P k 1
4
is true whenever
Pk is true.
Hence, by the principle of mathematical induction, statement
numbers i.e., N.
Pn
is true for all natural
Question 5:
Prove the following by using the principle of mathematical induction for all n
N:
n1
n
2 3 n
2 1 3 3
1.3 2.3 3.3 ..... n.3
4
Solution 5:
Let the given statement be
,
P n
i.e.,
n1
n
2 3 n
2 1 3 3
Pn
:1.3 2.3 3.3 .... n3
4
For
n 1, we have
11
2
2.11 3 3 3 3 12
P1 :1.3 3 3
4 4 4
Let
Pk be true for some positive integer k, i.e.,
k 1
2 3 k
2k
1 3 3
1.3 2.3 3.3 ..... k3
....... i
4
We shall now prove that
Consider
P k 1
1.3 2.3 3.3 ..... k.3 k
1 .3
is true.
2 3 k
k
1
, which is true.