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Byju's Answer
Standard XII
Mathematics
Sum of n Terms
Let sn be t...
Question
Let
s
n
be the sum of all integers k such that
2
n
<
k
<
2
n
+
1
,
for
n
≥
1.
Then
9
divides
S
n
, if and only if
A
n
is odd
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B
n
is of the form
3
k
+
1
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C
n
is even
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D
n
is of the form
3
k
+
2
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Solution
The correct option is
C
n
is even
s
n
=
(
2
n
+
1
)
+
(
2
n
+
2
)
+
(
2
n
+
3
)
+
.
.
.
.
.
.
.
+
(
2
n
+
2
n
−
1
)
s
n
=
(
2
n
−
1
)
×
(
2
n
)
+
(
1
+
2
+
3......
+
(
2
n
−
1
)
)
s
n
=
(
2
n
−
1
)
×
(
2
n
)
+
(
2
n
−
1
)
×
(
2
n
)
2
s
n
=
(
2
n
−
1
)
×
3
×
2
n
−
1
For it to be divisible by
9
,
(
2
n
−
1
)
must bedivisible by
3
.
Therefore
n
must be even.
Suggest Corrections
0
Similar questions
Q.
Let
S
n
be the sum
of
all
integers k such that
2
n
<
k
<
2
n
+
1
,
for
n
≥
1
then 9 divides
S
n
if and only if
Q.
Let
S
n
be the sum of all integers k such that
2
n
<
k
<
2
n
−
1
, for n > 1, Then
9
divides
S
n
if and only if
Q.
Let
S
n
=
1
2
n
+
1
√
4
n
2
−
1
+
1
√
4
n
2
−
4
+
.
.
.
.
.
.
.
.
+
1
√
3
n
2
+
2
n
−
1
,
n
∈
N
, if
lim
n
→
∞
S
n
=
α
then which of the following is defined
Q.
Let
S
n
=
n
(
n
+
1
)
(
n
+
2
)
+
n
(
n
+
2
)
(
n
+
4
)
+
n
(
n
+
3
)
(
n
+
6
)
+
.
.
.
.
.
.
+
1
6
n
, then
lim
n
→
∞
S
n
is
Q.
If S
n
=
∑
r
=
1
n
1
+
2
+
2
2
+
.
.
.
Sum
to
r
terms
2
r
, then S
n
is equal to
(a) 2
n
− n − 1
(b)
1
-
1
2
n
(c)
n
-
1
+
1
2
n
(d) 2
n
− 1
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