Question

show that the mean value of a complete G.C.cycle is zero.​

Answer 1

Answer:

The mean value or average value of AC over a complete cycle is zero. So we do not consider the average value of AC over one cycle. We therefore take the average value of AC over a half cycle only I.e. from t=0 to t=T/2

Answer 2

A complete G.C. (geometrically-centered) cycle consists of a sequence of numbers that are each multiplied by a fixed ratio, which is called the common ratio. Let the common ratio be denoted by r, and let the sequence start with the first term a. Then the G.C. cycle consists of the terms a, ar, ar^2, ar^3, ..., ar^(n-1), where n is the number of terms in the cycle.

What is Ratio ?

A ratio is a comparison of two or more identical quantities. It describes how the two quantities are related in terms of their size, worth, or quantity. Ratios can be written as a fraction, a decimal, or a percentage, among other formats.

To show that the mean value of a complete G.C. cycle is zero, we need to compute the sum of all the terms in the cycle and divide it by the number of terms. Since the cycle is complete, we know that the sum of all the terms is given by:

S = a + ar + ar^2 + ar^3 + ... + ar^(n-1)

We can rearrange this sum by factoring out the common ratio r:

S = a(1 + r + r^2 + r^3 + ... + r^(n-1))

The sum inside the parentheses is a geometric series, which has a well-known formula:

1 + r + r^2 + r^3 + ... + r^(n-1) = (r^n - 1) / (r - 1)

Substituting this formula into the expression for S, we get:

S = a((r^n - 1) / (r - 1))

Now, to find the mean value of the cycle, we divide S by the number of terms n:

mean = S / n

Substituting our expression for S, we get:

mean = a((r^n - 1) / (n(r - 1)))

Now, we can see that the mean value is zero if and only if the numerator of this expression is zero:

r^n - 1 = 0

This equation has n roots, which are the n-th roots of unity. These roots are evenly spaced around the unit circle in the complex plane, and they add up to zero because they form a closed loop. Therefore, we can conclude that the mean value of a complete G.C. cycle is zero.

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