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Byju's Answer
Standard XII
Mathematics
Proof by mathematical induction
72n + 23n - 3...
Question
7
2
n
+
2
3
n
−
3
.3
n
−
1
is divisible by
25
for any natural number
n
≥
1
. Prove that by mathematical
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Solution
Step 1. Check if
P
(
i
)
is correct
We get
P
(
1
)
=
7
2
+
2
0
×
3
0
=
49
+
1
=
50
Which is divisible by
25
So
P
(
1
)
is true.
Step 2. Let
P
(
K
)
is true, which means
7
2
k
+
3
3
k
−
3
3
k
−
1
=
25
m
When
m
=
i
n
t
e
g
e
r
Now check if
P
(
K
+
1
)
is True
Hence,
P
(
K
H
)
=
7
2
(
K
+
H
)
+
2
3
(
K
+
1
)
−
3
×
3
(
K
+
1
)
−
1
=
(
7
2
k
×
49
)
+
2
(
3
k
−
3
)
×
8
×
3
(
k
−
1
)
×
3
=
49
×
7
2
k
+
24
[
2
3
k
−
3
3
k
−
1
)
]
=
49
×
7
2
k
+
24
[
25
m
−
7
2
k
]
=
49.7
2
k
−
24.7
2
k
+
24.25
m
=
25
[
7
2
k
+
24
m
]
=
25
×
some integers value
∴
P
(
K
+
1
)
is true whenerver
P
(
K
)
is true.so using
P
M
I
, we can conclude
P
(
n
)
is true for all
n
∈
N
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