The value of 1+3+5+7+...........(2n-1) is
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Answered by
1
Step 1: Prove true for
n
=
1
LHS=
2
−
1
=
1
RHS=
1
2
=
1
= LHS
Therefore, true for
n
=
1
Step 2: Assume true for
n
=
k
, where k is an integer and greater than or equal to 1
1
+
3
+
5
+
7
+
...
.
+
(
2
k
−
1
)
=
k
2
------- (1)
Step3: When
n
=
k
+
1
,
RTP:
1
+
3
+
5
+
7
+
...
+
(
2
k
−
1
)
+
(
2
k
+
1
)
=
(
k
+
1
)
2
LHS:
1
+
3
+
5
+
7
+
...
+
(
2
k
−
1
)
+
(
2
k
+
1
)
=
k
2
+
(
2
k
+
1
)
---(from 1 by assumption)
=
(
k
+
1
)
2
=RHS
Therefore, true for
n
=
k
+
1
Step 4: By proof of mathematical induction, this statement is true for all integers greater than or equal to 1
(here, it actually depends on what your school tells you because different schools have different ways of setting out the final step but you get the gist of it)
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