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Joint Entrance Examination
Mathematics
Sequences and Series
Question 292
Question 292 of 519
Medium
MCQ
If
2
10
+
2
9
⋅
3
1
+
2
8
⋅
3
2
+
⋯
+
2
⋅
3
9
+
3
10
=
S
−
2
11
2^{10}+2^9\cdot3^1+2^8\cdot3^2+\cdots+2\cdot3^9+3^{10} = S-2^{11}
2
10
+
2
9
⋅
3
1
+
2
8
⋅
3
2
+
⋯
+
2
⋅
3
9
+
3
10
=
S
−
2
11
, then
S
S
S
is equal to:
A
3
11
−
2
12
3^{11}-2^{12}
3
11
−
2
12
B
3
11
3^{11}
3
11
C
3
11
2
+
2
10
\dfrac{3^{11}}{2}+2^{10}
2
3
11
+
2
10
D
2
⋅
3
11
2\cdot3^{11}
2
⋅
3
11
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Official solution
The LHS is a G.P.-type sum:
2
10
(
3
/
2
)
11
−
1
3
/
2
−
1
=
3
11
−
2
11
=
S
−
2
11
⟹
S
=
3
11
.
2^{10}\dfrac{(3/2)^{11}-1}{3/2-1} = 3^{11}-2^{11} = S-2^{11} \implies S=3^{11}.
2
10
3/2
−
1
(
3/2
)
11
−
1
=
3
11
−
2
11
=
S
−
2
11
⟹
S
=
3
11
.
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