In an A.P. the first 4 verses are x, p, y, 2p , what is the ratio of x and y? answer with explanation.
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Step-by-step explanation:
Let the first term and common difference of the first AP be a
1
and d
1
and the same for the second AP be a
2
and d
2
So,
n+1
2n
=
2
n
(2a
2
+(n−1)d
2
)
2
n
(2a
1
+(n−1)d
1
)
⇒
n+1
2n
=
2a
2
+(n−1)d
2
2a
1
+(n−1)d
1
⇒
n+1
2n
=
a
2
+(
2
n−1
)d
2
a
1
+(
2
n−1
)d
1
⇒
n+1
2n
=
a
2
+Nd
2
a
1
+Nd
1
Where N=
2
n−1
For the ratio of 8
th
terms N=7
Or
2
n−1
=7
⇒n−1=14 ⇒n=15
Thus,
a
2
+7d
2
a
1
+7d
1
=
15+1
2(15)
⇒
16
2(15)
=15:8
∴ the ratio of 8th term of the first and second AP is 15:8
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