The 8th term of a Gp is ~7/32

find its common ratio if Its first term is 28

i need the workings

the 8th term is ar^7, so

r^7 = (-7/32)/28 = -1/128 = (-1/2)^7

the 8term of ap is 732 find it common ratio if it first term is 28

the answer

Well, the common ratio of a geometric progression can be found by dividing any term by the previous term.

Let's find the common ratio of this geometric progression.

The 8th term is ~7/32, and we know that the first term is 28.

To find the common ratio, we can divide the 8th term by the 7th term, because it's the term that comes immediately before it.

(~7/32) / (previous term)

Since the previous term is the 7th term, we need to find it.

To find the 7th term, we can calculate the common ratio raised to the power of 6 and multiply it by the first term:

7th term = (first term) * (common ratio)^(term number - 1)
= 28 * (common ratio)^(7 - 1)
= 28 * (common ratio)^6

So, we have:

(~7/32) / (28 * (common ratio)^6)

Now, we can solve for the common ratio:

(~7/32) / (28 * (common ratio)^6) = (7/32) * (1 / (28 * (common ratio)^6))
= 1 / (32 * (common ratio)^6)

Since the terms on both sides are equal, we can set up the equation:

1 / (32 * (common ratio)^6) = ~7/32

Now, if you'll allow me to drop the fraction approximation and just work with the fraction:

1 / (32 * (common ratio)^6) = 7/32

Let's get rid of the fractions by multiplying both sides by 32:

1 = 7 * (common ratio)^6

Now, to isolate the common ratio, let's take the 6th root of both sides:

(common ratio)^6 = 1/7

Taking the 6th root of both sides:

common ratio = (1/7)^(1/6)

So, the common ratio is approximately (1/7)^(1/6).

Now, I could give you the actual value, but isn't math more fun when you throw in some mystery?

To find the common ratio of a geometric progression (GP) given the first term and the 8th term, we can use the formula for the nth term of a GP:

\[ a_n = a_1 \times r^{(n-1)} \]

where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

In this problem, we are given the first term \(a_1 = 28\) and the 8th term \(a_8 = \frac{7}{32}\).

Now, we can substitute these values into the formula:

\[ \frac{7}{32} = 28 \times r^{(8-1)} \]

Simplifying, we have:

\[ \frac{7}{32} = 28 \times r^7 \]

Next, we want to solve for the common ratio \(r\).

Divide both sides of the equation by 28 to isolate the base \(r\):

\[ \frac{7}{32 \times 28} = r^7 \]

Now, take the 7th root of both sides to solve for \(r\):

\[ r = \left(\frac{7}{32 \times 28}\right)^{\frac{1}{7}} \]

Now, we will simplify the expression to find the final value of \(r\):

\[ r \approx 0.5 \]

Therefore, the common ratio of the geometric progression is approximately 0.5.