the first term of arithmetic progression is -8 .and the ratio of the 7th and the 9th term is 5 :8 .find the common difference.

term7 = a+6d = 6d-8

term9 = a+8d = 8d-8

(6d-8)/(8d-8) = 5/8
48d - 64 = 40d - 40
8d = 24
d = 3

To find the common difference of an arithmetic progression, we need to know at least two terms in the progression. In this case, we have the first term (-8), so we need to find another term.

We are also given the ratio of the 7th term to the 9th term, which is 5:8. Let's first find the 7th term and the 9th term.

Since the common difference is constant in an arithmetic progression, we can use the formula to find the terms. The formula for the nth term of an arithmetic progression is:

nth term = first term + (n - 1) * common difference

Substituting the given values for the first term and the 7th term:

7th term = -8 + (7 - 1) * common difference
= -8 + 6 * common difference
= -8 + 6d

9th term = -8 + (9 - 1) * common difference
= -8 + 8 * common difference
= -8 + 8d

Now, we are given that the ratio of the 7th term to the 9th term is 5 : 8:

(7th term) / (9th term) = 5 / 8

Replacing the expressions for the 7th and 9th term:

(-8 + 6d) / (-8 + 8d) = 5 / 8

Now, we can solve this equation to find the common difference:

Multiply both sides by (8 - 8d) to eliminate the denominators:

(-8 + 6d) * (8 - 8d) = 5 * (-8 + 8d)

Expand the equation:

-64 + 64d + 48d - 48d^2 = -40 + 40d

Rearrange the terms:

-48d^2 + 112d - 64 = -40 + 40d

Combine like terms:

-48d^2 + 112d + 40d - 64 + 40 = 0

Simplify:

-48d^2 + 152d - 24 = 0

Divide the equation by -8 to simplify further:

6d^2 - 19d + 3 = 0

Now we need to solve this quadratic equation for d. Since this equation cannot be easily factorized, we'll use the quadratic formula:

d = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 6, b = -19, and c = 3. We can substitute these values:

d = (-(-19) ± √((-19)^2 - 4 * 6 * 3)) / (2 * 6)
d = (19 ± √(361 - 72)) / 12
d = (19 ± √289) / 12
d = (19 ± 17) / 12

Now we'll find the two possible values of d by considering both the positive and negative square root:

1. (19 + 17) / 12 = 36 / 12 = 3

2. (19 - 17) / 12 = 2 / 12 = 1/6

Therefore, the common difference can be either 3 or 1/6.