In a sequence of (4n + 1) terms, the first (2n+1) terms are in A.P., whose common difference is 2 and the last 2n + 1 terms are in G.P., whose common ratio is 0.5. If the middle term of an A.P. and G.P. are equal then the middle term of the sequence is -
Text Solution
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First (2n + 1) terms of an A.P.
a, a + 2, a + 4 …. a + 2.2n
The middle term of the sequence 4n + 1 terms is (2n + 1) th term i.e. (a + 4n)
The middle term of the A.P. of 2n + 1 terms is (n + 1) th term i.e. a + 2n
Again the last 2n + 1 terms the first term will be (2n + 1) th term of the A.P. i.e. a + 4n
∴ G.P. is (a + 4n), (a + 4n) (0.5) , (a + 4n) (0.5) 2 … (a + 4n) (0.5) 2n
Middle term is (a + 4n) (0.5) n
According to given conditions
a + 2n = (a + 4n) (0.5) n
a = 
Required middle term is a + 4n
⇒ 
+ 4n
⇒ 
⇒ 
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