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Stuti Srivastava
Subject: Maths
, asked on 20/5/18
How to break (k^3 +6k^2+9k+4) this such that it becomes (k+1)(k+1)(k+4).
Answer
1
Sanitya Srivastava
Subject: Maths
, asked on 28/4/18
Experts kindly help
Answer
3
Saswat Das
Subject: Maths
, asked on 14/4/18
Q.
Find the differentiation
1
.
y
=
sin
2
x
-
cos
2
x
2
.
y
=
cos
2
x
+
1
3
3
.
y
=
cos
x
3
+
sin
2
x
4
.
y
=
3
x
2
-
9
2
sin
x
Answer
1
Drishti
Subject: Maths
, asked on 23/3/18
S
h
o
w
t
h
a
t
:
-
1
1
1
+
3
x
1
+
3
y
1
1
1
1
+
3
x
1
=
9
(
3
x
y
z
+
x
y
+
y
x
+
z
x
)
Answer
2
Ridhit Jain
Subject: Maths
, asked on 6/3/18
Prove by PMI
3
.
2
2
+
3
2
.
2
3
+
.
.
.
.
.
.
.
.
.
.
.
+
3
n
2
n
+
1
=
12
5
6
n
-
1
prove
Answer
1
Kanishk Sharma
Subject: Maths
, asked on 27/2/18
Q.4. Prove that
3
3
n
-
26
n
-
1
is divisible by 676.
Answer
1
Dharunika Vijayakumar
Subject: Maths
, asked on 14/2/18
TELL ME THE ANSWER FOR 15th QUESTION . PLEASE TELL ME FAST
Q15. Prove that ( cos
θ
+
i
sin
θ
)
n
= cos
n
θ
+
i
sin
(
n
θ
)
, for all n
∈
N by using PMI.
Answer
3
Tanmaya Darisi
Subject: Maths
, asked on 8/2/18
Using PMI prove that 1 × 1! + 2 × 2! + 3 × 3! + -----+ n × n! = (n + 1)! – 1 for all n∈N
Answer
1
Naga Nandhini
Subject: Maths
, asked on 31/1/18
Experts,explain the one underlined after
Answer
1
Naga Nandhini
Subject: Maths
, asked on 31/1/18
prove that 1+2+3+....+n=n(n+1)/2 by pmi
Answer
1
Ashwini Upadhya
Subject: Maths
, asked on 20/1/18
Please solve it using mathematical induction
P
r
o
v
e
t
h
a
t
(
1
+
x
)
n
≥
1
+
n
x
,
f
o
r
a
l
l
n
a
t
u
r
a
l
n
u
m
b
e
r
n
,
w
h
e
r
e
x
>
-
1
b
y
m
a
t
h
e
m
a
t
i
c
a
l
i
n
d
u
c
t
i
o
n
.
Answer
2
Ranjan Shivam
Subject: Maths
, asked on 2/1/18
Q. if a, b, c, are in A.P. , then
1
b
+
c
,
1
c
+
a
,
1
a
+
b
a
r
e
i
n
(
A
)
A
.
P
.
(
B
)
G
.
P
.
(
C
)
H
.
P
.
(
D
)
N
o
n
e
o
f
t
h
e
s
e
Answer
1
Nura Mohammad
Subject: Maths
, asked on 25/11/17
Q. Prove that
2
n
>
n
f
o
r
a
l
l
p
o
s
i
t
i
v
e
i
n
t
e
g
e
r
s
n
.
Please explain the last line of solution
Answer
1
Nura Mohammad
Subject: Maths
, asked on 25/11/17
How is the underlined step coming?
Answer
1
Nura Mohammad
Subject: Maths
, asked on 25/11/17
Prove the following by the principle of mathematical induction.
2n?>?n2, where?n?is a positive integer such that?n?> 4.
Solution:
Let the given statement be P(n), i.e.,
P(n) : 2n?>?n2?where?n?> 4
For?n?= 5,
25?= 32 and 52?= 25
?25?> 52
Thus, P(n) is true for?n?= 5.
Let P(n) be true for?n?=?k, i.e.,
2k?>?k2?? (1)
Now, we have to prove that P(k? 1) is true whenever P(k) is true, i.e. we have to prove that 2k? 1?> (k? 1)2.
From equation (1), we obtain
2k?>?k2
On multiplying both sides with 2, we obtain
2 ? 2k?> 2 ??k2
2k? 1?> 2k2
?To prove 2k? 1?> (k? 1)2, we only need to prove that 2k2?> (k? 1)2.
Let us assume 2k2?> (k? 1)2.
? 2k2?>?k2? 2k? 1
??k2?> 2k? 1
??k2?? 2k?? 1 > 0
? (k?? 1)2?? 2 > 0
? (k?? 1)2?> 2, which is true as?k?> 4
Hence, our assumption 2k2?> (k? 1)2?is correct and we have 2k? 1?> (k? 1)2.
Thus, P(n) is true for?n?=?k? 1.
Thus, by the principle of mathematical induction, the given mathematical statement is true for every positive integer?n.
?
IN THIS EXAMPLE IT HAS BEEN WRITTEN THAT ,
To prove 2k? 1?> (k? 1)2, we only need to prove that 2k2?> (k? 1)2.
how?2k? 1?> (k? 1)2?=?2k2?> (k? 1)2
I cant understand how so plz explain that
Answer
1
1
2
3
4
5
Next
What are you looking for?
Q. Find the differentiation
prove
Q15. Prove that ( cos )n = cos , for all n N by using PMI.
Please explain the last line of solution
2n?>?n2, where?n?is a positive integer such that?n?> 4.
Solution:
Let the given statement be P(n), i.e.,
P(n) : 2n?>?n2?where?n?> 4
For?n?= 5,
25?= 32 and 52?= 25
?25?> 52
Thus, P(n) is true for?n?= 5.
Let P(n) be true for?n?=?k, i.e.,
2k?>?k2?? (1)
Now, we have to prove that P(k? 1) is true whenever P(k) is true, i.e. we have to prove that 2k? 1?> (k? 1)2.
From equation (1), we obtain
2k?>?k2
On multiplying both sides with 2, we obtain
2 ? 2k?> 2 ??k2
2k? 1?> 2k2
?To prove 2k? 1?> (k? 1)2, we only need to prove that 2k2?> (k? 1)2.
Let us assume 2k2?> (k? 1)2.
? 2k2?>?k2? 2k? 1
??k2?> 2k? 1
??k2?? 2k?? 1 > 0
? (k?? 1)2?? 2 > 0
? (k?? 1)2?> 2, which is true as?k?> 4
Hence, our assumption 2k2?> (k? 1)2?is correct and we have 2k? 1?> (k? 1)2.
Thus, P(n) is true for?n?=?k? 1.
Thus, by the principle of mathematical induction, the given mathematical statement is true for every positive integer?n.
?
IN THIS EXAMPLE IT HAS BEEN WRITTEN THAT ,
To prove 2k? 1?> (k? 1)2, we only need to prove that 2k2?> (k? 1)2.
how?2k? 1?> (k? 1)2?=?2k2?> (k? 1)2
I cant understand how so plz explain that