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Byju's Answer
Standard XII
Mathematics
Proof by mathematical induction
13.7+17.11+11...
Question
1
3
.
7
+
1
7
.
11
+
1
11
.
5
+
.
.
.
+
1
(
4
n
-
1
)
(
4
n
+
3
)
=
n
3
(
4
n
+
3
)
Open in App
Solution
Let P(n) be the given statement.
Now,
P
(
n
)
=
1
3
.
7
+
1
7
.
11
+
1
11
.
15
+
.
.
.
+
1
(
4
n
-
1
)
(
4
n
+
3
)
=
n
3
(
4
n
+
3
)
Step
1
:
P
(
1
)
=
1
3
.
7
=
1
21
=
1
3
(
4
+
3
)
Hence
,
P
(
1
)
i
s
t
r
u
e
.
Step
2
:
Let
P
(
m
)
is
true
.
Then
,
1
3
.
7
+
1
7
.
11
+
.
.
.
+
1
(
4
m
-
1
)
(
4
m
+
3
)
=
m
3
(
4
m
+
3
)
T
o
prove
:
P
(
m
+
1
)
is
true
.
T
h
a
t
i
s
,
1
3
.
7
+
1
7
.
11
+
.
.
.
+
1
(
4
m
+
3
)
(
4
m
+
7
)
=
m
+
1
3
(
4
m
+
7
)
Now
,
P
(
m
)
=
1
3
.
7
+
1
7
.
11
+
.
.
.
+
1
(
4
m
-
1
)
(
4
m
+
3
)
=
m
3
(
4
m
+
3
)
⇒
1
3
.
7
+
1
7
.
11
+
.
.
.
+
1
(
4
m
-
1
)
(
4
m
+
3
)
+
1
(
4
m
+
3
)
(
4
m
+
7
)
=
m
3
(
4
m
+
3
)
+
1
(
4
m
+
3
)
(
4
m
+
7
)
Adding
1
(
4
m
+
3
)
(
4
m
+
7
)
to
both
sides
⇒
1
3
.
7
+
1
7
.
11
+
.
.
.
+
1
(
4
m
+
3
)
(
4
m
+
7
)
=
4
m
2
+
7
m
+
3
3
(
4
m
+
3
)
(
4
m
+
7
)
=
(
4
m
+
3
)
(
m
+
1
)
3
(
4
m
+
3
)
(
4
m
+
7
)
=
m
+
1
3
(
4
m
+
7
)
Thus
,
P
(
m
+
1
)
is
true
.
B
y
t
h
e
p
rinciple
of
m
athematical
i
nduction
,
P
(
n
)
is
true
for
all
n
∈
N
.
Suggest Corrections
0
Similar questions
Q.
Evaluate :
(
−
√
−
1
)
4
n
+
3
,
n
∈
N
Q.
(
−
√
−
1
)
4
n
+
3
(n, +ive integer)
Q.
The value of
-
-
1
4
n
-
3
,
where n ∈ N, is ____________.
Q.
1
3.7
+
1
7.11
+
1
11.15
+
.
.
.
.
+
1
(
4
n
−
1
)
(
4
n
+
3
)
=
n
3
(
4
n
+
3
)
Q.
Show that
(
−
√
−
1
)
4
n
+
3
=
i
, where n is a positive integer.
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