Question

If 8; 2x; 2y forms an arithmetic sequence and 2x ; 2y; 36 forms a geometric sequence determine the values of x and y. Arithmetic and Geometric Series​

Answer 1

36

Explanation:

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric sequence is a sequence of numbers in which any two consecutive terms have a constant ratio.

Given that 8, 2x, 2y is an arithmetic sequence, we know that:

2x - 8 = 2y - 2x

x = y

Given that 2x, 2y, 36 is a geometric sequence, we know that:

(2y)/(2x) = 36/2y

y^2 = 36x

Now, we can use the first equation to substitute for y in the second equation, and get:

x^2 = 36x

x^2 - 36x = 0

x(x-36) = 0

So x = 0 or x = 36

However, x cannot be equal to zero because in that case the second term of the arithmetic sequence will be zero, and it will not be a valid solution. Therefore, x = 36.

So, y = x = 36.

Therefore, the values of x and y are 36.