The third term of a GP is 10, and the 7th term is 6250. What is the common ratio?

a r^2 = 10

a r^6 = 6250

(10/r^2)r^6 = 6250

r^4 = 625 = 25*25 = 5*5*5*5
so
r = 5

Yes

To find the common ratio (r) of a geometric progression (GP), we need to use the formula:

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

Where:
- \(a_n\) represents the \(n\)th term of the GP
- \(a_1\) represents the first term of the GP
- \(r\) represents the common ratio
- \(n\) is the position of the term in the GP

Given that the third term (n = 3) is 10 and the seventh term (n = 7) is 6250, we can form two equations using the formula above:

For the third term:
\[10 = a_1 \times r^{(3-1)}\] ---- (Equation 1)

For the seventh term:
\[6250 = a_1 \times r^{(7-1)}\] ---- (Equation 2)

To find the value of the common ratio (r), we can divide Equation 2 by Equation 1:

\[\frac{6250}{10} = \frac{a_1 \times r^{(7-1)}}{a_1 \times r^{(3-1)}}\]

Simplifying the equation:

\[625 = r^{(7-3)}\]

\[625 = r^4\]

To find the value of \(r\), we take the fourth root of both sides of the equation:

\[\sqrt[4]{625} = \sqrt[4]{r^4}\]

\[5 = r\]

So, the common ratio (r) is 5.

To find the common ratio (r) of a geometric progression (GP), we need to use the formula for the nth term of a GP:

Tn = a * r^(n-1)

Where:
Tn = nth term of the GP
a = first term of the GP
r = common ratio of the GP
n = position of the term in the GP

Given that the third term is 10, we can substitute the values into the formula as follows:

10 = a * r^(3-1)
10 = a * r^2 ---(Equation 1)

Similarly, the 7th term is given as 6250, so we have:

6250 = a * r^(7-1)
6250 = a * r^6 ---(Equation 2)

Now, we have a system of two equations (Equation 1 and Equation 2) with two unknowns (a and r). We can solve these equations simultaneously to find the values of a and r.

First, we'll take Equation 1 and divide it by Equation 2 to eliminate the 'a' term:

(10 / 6250) = (a * r^2) / (a * r^6)

Simplifying further:

1/625 = 1/r^4
r^4 = 625

To solve for r, we take the fourth root of both sides:

r = ∛(625)
r = 5

Therefore, the common ratio (r) of the geometric progression is 5.