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Exercise 4.1
Prove the following by using the principle of mathematical induction for all N:
1.
Ans. Let
For
1 = 1
is true.
Now, let be true for
……….(i)
For [Using eq. (i)]
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
2.
Ans. Let
For
1 = 1
is true.
Now, let be true for
……….(i)
For
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
3.
Ans. Let
For
1 = 1
is true.
Now, let be true for
……….(i)
For
[Using (i)]
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
4.
Ans. Let
For
6 = 6
is true.
Now, let be true for
………(i)
For
[Using eq. (i)]
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
5.
Ans. Let
For
3 = 3
is true.
Now, let be true for
For
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
6.
Ans. Let
For
2 = 2
is true.
Now, let be true for
………(i)
For
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
7.
Ans. Let
For
3 = 3
is true.
Now, let be true for
For
=
=
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
8.
Ans. Let
For
2 = 2
is true.
Now, let be true for
For
=
= =
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
9.
Ans. Let
For
is true.
Now, let be true for
For
= =
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
10.
Ans. Let
For
is true.
Now, let be true for
For
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
11.
Ans. Let
For
is true.
Now, let be true for
……….(i)
For
R.H.S. =
And L.H.S. = [Using eq. (i)]
=
=
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
12.
Ans. Let
For
is true.
Now, let be true for
……….(i)
For
R.H.S. =
L.H.S. =
[Using eq. (i)]
L.H.S. =
=
=
=
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
13.
Ans. Let
For
is true.
Now, let be true for
……….(i)
For
R.H.S. =
L.H.S. =
[Using eq. (i)]
L.H.S. =
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
14.
Ans. Let
For
is true.
Now, let be true for
For R.H.S. =
L.H.S. =
[Using eq. (i)]
L.H.S. = =
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
15.
Ans. Let
For
is true.
Now, let be true for
……….(i)
For R.H.S. =
L.H.S. =
[Using eq. (i)]
=
=
=
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
16.
Ans. Let
For
is true.
Now, let be true for
……….(i)
For R.H.S. =
L.H.S. =
L.H.S. =
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
17.
Ans. Let
For
is true.
Now, let be true for
……….(i)
For R.H.S. =
L.H.S. =
L.H.S. =
=
=
=
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
18.
Ans. Let
For
is true.
Now, let be true for
……….(i)
For ,
Now, adding on both sides of eq. (i), we have
8 < 9
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
19. is a multiple of 3.
Ans. Let is a multiple of 3.
For 1 (1 + 1) (1 + 5) is a multiple of 3 = 12 is a multiple of 3
P (1) is true.
Let be true for
,
is a multiple of 3.
….(i)
For ,
is a multiple of 3
Now,
=
= [Using (i)]
=
=
= is a multiple of 3
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
20. is divisible by 11.
Ans. Let is divisible by 11.
For is divisible by 11
= 11 is divisible by 11
P (1) is true.
Let be true for
,
is divisible by 11 =
……….(i)
For is divisible by 11
is divisible by 11
Now,
=
=
is divisible by 11
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
21. is divisible by
Ans. Let is divisible by
For is divisible by
=
is divisible by
P (1) is true.
Let be true for
,
is divisible by
=
……….(i)
For is divisible by
Now,
=
=
= [From eq. (i)]
=
is divisible by
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
22. is divisible by 8.
Ans. Let is divisible by 8.
For is divisible by 8 = 64 is divisible by 8
P (1) is true.
Let be true for
,
is divisible by 8 =
……….(i)
For is divisible by 8
is divisible by 8
Now,
= [From eq. (i)]
=
= =
is divisible by 8
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
23. is a multiple of 27.
Ans. Let is a multiple of 27.
For is a multiple of 27 = 27 is a multiple of 27
P (1) is true.
Let be true for
,
is a multiple of 27 =
…..(i)
For is a multiple of 27
Now,
=
=
= [From eq. (i)]
=
is a multiple of 27
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
24.
Ans. Let
For
9 < 16
P (1) is true.
Let be true for
……….(i)
For
=
Now, adding 2 on both sides in eq. (i),
Also
is true.
Therefore, is true.
is true.
Hence by Principle of Mathematical Induction, is true for all
N.
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