the average of first four numbers in a GP is 140. find the ratio.

To find the average of the first four numbers in a geometric progression (GP), we need to know the first term and the common ratio.

Let's represent the first term as "a" and the common ratio as "r".

The formula for the sum of the first four terms in a GP is:

Sum = a + ar + ar^2 + ar^3

Since we want the average, we divide the sum by 4:

Average = (a + ar + ar^2 + ar^3) / 4

Given that the average is 140, we have:

140 = (a + ar + ar^2 + ar^3) / 4

Now, we can solve for the common ratio (r).

Multiplying both sides of the equation by 4:

560 = a + ar + ar^2 + ar^3

Rearranging the terms:

ar^3 + ar^2 + ar + a = 560

Factoring out "a":

a(r^3 + r^2 + r + 1) = 560

Since "a" cannot be equal to zero, we can divide both sides of the equation by "a":

r^3 + r^2 + r + 1 = 560 / a

Since we are looking for an integer ratio, we need to find the value of "a" that makes the right side of the equation an integer.

Given that the right side should be an integer, we can try different values for "a" and see which one works. Let's start by assuming "a" is equal to 1:

r^3 + r^2 + r + 1 = 560 / 1

r^3 + r^2 + r + 1 = 560

We can use trial and error, or use a numerical method, to find the approximate value of "r". By trying different values, we find that "r" is approximately 2.

To confirm, let's substitute "r" = 2 into the equation:

2^3 + 2^2 + 2 + 1 = 8 + 4 + 2 + 1 = 15

Since 15 is not equal to 560, we can conclude that assuming "a" = 1 does not give us an integer ratio.

We need to try different values for "a" until we find one that satisfies the equation "r^3 + r^2 + r + 1 = 560 / a".

Unfortunately, there isn't a unique, single solution to this equation. The ratio (r) depends on the choice of the first term (a). You would need to specify additional information or constraints to find a unique solution to this problem.