In a sequence of 4n+1 terms the first 2n+1 terms are in Ap whose common difference is 2,and the last 2n+1 terms are in Gp whose common ratio is 0.5 If the middle terms of the Ap and the Gp are equal then the middle term of sequence is Ans n×2to the power n+1/2to power n and -1
The midpoint of a sequence of length (2n+1) is n steps away from either endpoint. Suppose x is the value at the midpoint of the whole sequence. That's what you want to find.
The midpoint of the AP is n steps to the left of overall midpoint, making the midpoint of the AP equal to x - 2n.
The midpoint of the GP is n steps to the right of the overall midpoint, making that value (0.5^n)x.
Those two values are equal, so:
x - 2n = (0.5^n)x
x - (0.5^n)x = 2n
x(1 - 0.5^n) = 2n
x = 2n / (1 - 0.5^n)
You can get rid of the decimal fraction by multiplying that by 2^n/2^n:
x = n * 2^(n+1) / (2^n - 1)
PS: You should add information to the question. Look for an Update or Additional Information button. There's something like that. To add extensive details where you run out of room in the question plus an update or two, you can also "answer" your own question, putting the details there. You have more room in answers than you do in questions.
The midpoint of the AP is n steps to the left of overall midpoint, making the midpoint of the AP equal to x - 2n.
The midpoint of the GP is n steps to the right of the overall midpoint, making that value (0.5^n)x.
Those two values are equal, so:
x - 2n = (0.5^n)x
x - (0.5^n)x = 2n
x(1 - 0.5^n) = 2n
x = 2n / (1 - 0.5^n)
You can get rid of the decimal fraction by multiplying that by 2^n/2^n:
x = n * 2^(n+1) / (2^n - 1)
PS: You should add information to the question. Look for an Update or Additional Information button. There's something like that. To add extensive details where you run out of room in the question plus an update or two, you can also "answer" your own question, putting the details there. You have more room in answers than you do in questions.