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Byju's Answer
Standard XII
Mathematics
Inequalities Involving Mathematical Means
11n+2 + 122n+...
Question
11
n
+2
+ 12
2n
+1
is divisible by 133 for all n ∈ N.
Open in App
Solution
Let P(n) be the given statement.
Now,
P
n
:
11
n
+
2
+
12
2
n
+
1
is
divisible
by
133
.
Step
1
:
P
1
=
11
1
+
2
+
12
2
+
1
=
1331
+
1728
=
3059
I
t
is
divisible
by
133
.
Step
2
:
Let
P
m
be
divisible
by
133
.
N
o
w
,
11
m
+
2
+
12
2
m
+
1
is
divisible
by
133
.
Suppose
:
11
m
+
2
+
12
2
m
+
1
=
133
λ
.
.
.
(
1
)
We
shall
show
that
P
m
+
1
is
true
whenever
P
m
is
true
.
Now
,
P
m
+
1
=
11
m
+
3
+
12
2
m
+
3
=
11
m
+
2
.
11
+
12
2
m
+
1
.
12
2
+
11
.
12
2
m
+
1
-
11
.
12
2
m
+
1
=
11
11
m
+
2
+
12
2
m
+
1
+
12
2
m
+
1
144
-
11
=
11
.
133
λ
+
12
2
m
+
1
.
133
From
(
1
)
=
133
11
λ
+
12
2
m
+
1
I
t
is
divisible
by
133
.
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.
11
n
+
2
+
12
2
n
+
1
is divisible by 133 for all
n
ϵ
N
.
Q.
Prove that
11
n
+
2
+
12
2
n
+
1
is divisible by
133
for any non-negative integral
n
.
Q.
If n ∈ N, then
11
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+
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+
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+
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is divisible by
Q.
Explain the method of mathematical induction and use it to show that
11
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+
2
+
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n
+
1
where n is natural number is divisible by 133
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