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Question
Mathematics
Let a 1= b 1=1, a n = a n -1+2 and bn=an+bn-1 for every natural number n ≥ 2. Then displaystyle∑n=115 an ⋅ bn is equal to
Q. Let
a
1
=
b
1
=
1
,
a
n
=
a
n
−
1
+
2
and
b
n
=
a
n
+
b
n
−
1
for every natural number
n
≥
2
.
Then
n
=
1
∑
15
a
n
⋅
b
n
is equal to ______
207
147
JEE Main
JEE Main 2022
Sequences and Series
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Answer:
27560
Solution:
a
1
=
b
1
=
1
a
2
=
a
1
+
2
=
3
a
3
=
a
2
+
2
=
5
a
4
=
a
2
+
2
=
7
⇒
a
n
=
2
n
−
1
b
2
=
a
1
+
b
1
=
4
b
3
=
a
3
+
b
2
=
9
b
4
=
a
4
+
b
3
=
16
b
n
=
n
2
n
=
1
∑
15
a
n
b
n
n
=
1
∑
15
(
2
n
−
1
)
n
2
n
=
1
∑
15
(
2
n
3
−
n
2
)
=
2
4
n
2
(
n
+
1
)
2
−
6
n
(
n
+
1
)
(
2
n
+
1
)
Put
n
=
15
=
4
2
×
225
×
16
×
16
−
6
15
×
16
×
31
=
27560