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Byju's Answer
Standard XII
Mathematics
Composite Function
If fx represe...
Question
If f(x) represents the least integer greater than
x
then prove that
(
√
3
+
1
)
2
n
n
∈
N
is divisible by
2
n
Open in App
Solution
(
√
3
+
1
)
2
n
=
(
(
1
+
√
3
)
2
)
n
=
(
1
+
3
+
2
√
3
)
n
=
(
4
+
2
√
3
)
n
=
2
n
(
2
+
√
3
)
n
=
2
n
(
2
n
+
n
C
1
2
n
−
1
√
3
1
+
n
C
2
2
n
−
2
√
3
2
+
n
C
3
2
n
−
3
√
3
3
.
.
.
+
n
C
n
√
3
n
)
=
2
n
(
k
)
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0
Similar questions
Q.
Assertion :If
n
is an odd integer greater than 3 but not a multiple of 3, then
(
x
+
1
)
n
−
x
n
−
1
is divisible by
x
3
+
x
2
+
x
Reason: If
n
is an odd integer greater than
3
but not a multiple of
3
, we have
(
1
+
ω
n
+
ω
2
n
)
=
0
Q.
Statement 1: If n is an odd integer greater than 3 but not a multiple of 3, then
(
x
+
1
)
n
−
x
n
−
1
is divisible by
x
3
+
x
2
+
x
.
Statement 2: If n is an odd integer greater than 3 but not a multiple of 3, we have
1
+
ω
n
+
ω
2
n
=
3
.
Q.
Prove that
1
−
2
n
+
2
n
(
2
n
−
1
)
2
!
−
2
n
(
2
n
−
1
)
(
2
n
−
1
)
3
!
+
.
.
.
+
(
−
1
)
n
−
1
2
n
(
n
−
1
)
.
.
.
(
n
+
2
)
(
n
−
1
)
=
(
−
1
)
n
+
1
(
2
n
)
2
(
n
!
)
2
,
where n is a + ive integer.
Q.
Prove that x
(
x
n
−
1
−
n
a
n
−
1
)
+
a
n
(
n
−
1
)
is divisible by
(
x
−
a
)
2
for all positive integers n greater than 1
Q.
Prove that
(
⌊
33
)
is divisible by
2
16
. What is the largest integer
n
such that
(
⌊
33
)
is divisible by
2
n
?
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