Exercise 10.1
Page no -10.16
Type 1
Question 1
Let $P(n)$ be the statement ' $1^{2}+2^{2}+3^{2}+\ldots+n^{2}=\frac{n(n+1)(2 n+1) \text { ' }}{6}$. Show that $P(1), P(2)$ and $P(3)$ are true.
Question 2
If $P(n)$ be the statement '10nt3 is a prime number', then prove that $P(1)$ $P(2)$ are true but $P(3)$ is false.
Question 3
If $P(n)$ be the statement ' $2^{n}>n$ ' and if $P(m)$ is true, show that $P(m+1)$ also true.
Prove the following by using the principle of mathematical induction :
Question 4
$4+8+12+\ldots+4 n=2 n(n+1)$ for all $n \in N$.
Question 5
$1+2+3+\ldots+n=\frac{n(n+1)}{2}$
Question 6
$1^{2}+2^{2}+3^{2}+\ldots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$
Question 7
$1+3+3^{2}+\ldots+3^{n-1}=\frac{3^{n}-1}{2}$
Question 8
$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$
Question 9
$(a b)^{n}=a^{n} b^{n}$
Question 10
$1^{2}+3^{2}+5^{2}+\ldots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3}$
Question 11
$1^{3}+3^{3}+5^{3}+\ldots+(2 n-1)^{3}=n^{2}\left(2 n^{2}-1\right)$
Question 12
$3 \cdot 2^{2}+3^{2} \cdot 2^{3}+\ldots+3^{n} \cdot 2^{n+1}=\frac{12}{5}\left(6^{n}-1\right)$
Question 13
$1.3+2 \cdot 3^{2}+3 \cdot 3^{3}+\ldots+n \cdot 3^{n}=\frac{(2 n-1) 3^{n+1}+3}{4}$
Question 14
$a+a r+a r^{2}+\ldots+a r^{n-1}=\frac{a\left(1-r^{n}\right)}{1-r}, r \neq 1$
Question 15
$a+(a+d)+(a+2 d)+\ldots+[a+(n-1) d]=\frac{n}{2}[2 a+(n-1) d]$
Question 16
$\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\ldots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1}$
Question 17
$\frac{1}{3.7}+\frac{1}{7.11}+\frac{1}{11.15}+\ldots+\frac{1}{(4 n-1)(4 n+3)}=\frac{n}{3(4 n+3)}$
Question 18
$1.2+2.3+3.4+\ldots+n(n+1)=\frac{n(n+1)(n+2)}{3}$
Question 19
$1.3+3.5+5.7+\ldots+(2 n-1)(2 n+1)=\frac{n\left(4 n^{2}+6 n-1\right)}{3}$
Question 20
$\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\ldots+\frac{1}{(3 n-2)(3 n+1)}=\frac{n}{3 n+1}$
Page no -10.17
Question 21
$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.1}+\ldots+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{6 n+4}$
Question 22
$\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\ldots+\frac{1}{(2 n+1)(2 n+3)}=\frac{n}{3(2 n+3)}$
Question 23
$\frac{1}{1.2 .3}+\frac{1}{2.3 .4}+\frac{1}{3.4 .5}+\ldots+\frac{1}{n(n+1)(n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$
Question 24
$1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots+\frac{1}{1+2+3+\ldots+n}=\frac{2 n}{n+1}$
Question 25
$\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right) \ldots\left(1+\frac{1}{n}\right)=n+1$
Question 26
$\left(1+\frac{3}{1}\right)\left(1+\frac{5}{4}\right)\left(1+\frac{7}{9}\right) \ldots\left(1+\frac{2 n+1}{n^{2}}\right)=(n+1)^{2}$
Question 27
$(\cos \theta+i \sin \theta)^{n}=\cos n \theta+i \sin n \theta$
Question 28
$3.6+6.9+9.12+\ldots+3 n(3 n+3)=3 n(n+1)(n+2)$
Question 29
If $u_{0}=2, u_{1}=3$ and $u_{n+1}=3 u_{n}-2 u_{n-1}$, show that $u_{n}=2^{n}+1, n \in N$.
Question 30
If $a_{0}=0, a_{1}=1$ and $a_{n+1}=3 a_{n}-2 a_{n-1}$, prove that $a_{n}=2^{n}-1$
Question 31
If $A_{1}=\cos \theta, A_{2}=\cos 2 \theta$ and for every natural number $m>2$, $A_{m}=2 A_{m-1} \cos \theta-A_{m-2}$, prove that $A_{n}=\cos n \theta$.
Type 2
In $n$ is a natural number, using mathematical induction show that :
Question 32
32.$n(n+1)(n+5)$ is divisible by 6. 33. $n^{7}-n$ is a multiple of 7 .
Question 34
34.$4^{n}-3 n-1$ is divisible by 9
35. $9^{n}-8 n-1$ is divisible by 64 .
Question 36
$10^{2 n-1}+1$ is divisible by 11
Question 37
$x^{n}-y^{n}$ is divisible by $x+y$ when $n$ is an even integer.
Question 38
38.$2.7^{n}+3.5^{n}-5$ is divisible by $24
39. 3^{2 n+2}-8 n-9$ is divisible by 8
Question 40
40.$x^{2 n}-y^{2 n}$ is divisible by $x+y$
41. $7^{n}-3^{n}$ is divisible by 4 .
Question 42
$(41)^{n}-(14)^{n}$ is a multiple of 27
Type 3
Show by using mathematical induction that:
Question 43
(i) $2^{n}>n$
(ii) $2^{n}>n^{2}, n \geq 5, n \in N$.
Question 44
$1^{2}+2^{2}+\ldots+n^{2}>\frac{n^{3}}{3}$
Question 45
$\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2 n}>\frac{13}{24}, n>1, n \in N$
Question 46
$1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\ldots+\frac{1}{n^{2}}<2-\frac{1}{n}, n \geq 2, n \in N$.
Question 47
$1+2+3+\ldots+n<(2 n+1)^{2}$
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