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Byju's Answer
Standard XI
Mathematics
Complex Plane
1+3+5+…+2 n-1...
Question
1 + 3 + 5 + ... + (2n − 1) = n
2
i.e., the sum of first n odd natural numbers is n
2
.
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Solution
Let P(n) be the given statement.
Now,
P
(
n
)
=
1
+
3
+
5
+
.
.
.
+
(
2
n
-
1
)
=
n
2
Step
1
:
P
(
1
)
=
1
=
1
2
Hence
,
P
(
1
)
is
true
.
Step
2
:
Let
P
(
m
)
b
e
true
.
Then
,
1
+
3
+
5
+
.
.
.
+
(
2
m
-
1
)
=
m
2
T
o
prove
:
P
(
m
+
1
)
is
true
.
i
.
e
.
,
1
+
3
+
5
+
.
.
.
+
2
m
+
1
-
1
=
m
+
1
2
⇒
1
+
3
+
5
+
.
.
.
+
2
m
+
1
=
m
+
1
2
Now
,
w
e
h
a
v
e
:
1
+
3
+
5
+
.
.
.
+
(
2
m
-
1
)
=
m
2
⇒
1
+
3
+
.
.
.
+
(
2
m
-
1
)
+
(
2
m
+
1
)
=
m
2
+
2
m
+
1
Adding
2
m
+
1
to
both
sides
⇒
1
+
3
+
5
+
.
.
.
+
(
2
m
+
1
)
=
(
m
+
1
)
2
Hence
,
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
.
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Similar questions
Q.
The sum of first n odd natural numbers is
(a) 2n − 1
(b) 2n + 1
(c) n
2
(d) n
2
− 1
