For an A.P. the first term is 25 and 20th term is 500. Find the common difference.

a1 = 25

an = a1 + ( n- 1 ) d

a20 = a1 + ( 20 - 1 ) d

500 = 25 + 19 d Subtract 25 to both sides

500 - 25 = 25 + 19 d - 25

475 = 19 d Divide both sides by 19

475 / 19 = d

25 = d

d = 25

To find the common difference of an arithmetic progression (A.P.), we can use the formula:

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

Given that the first term is 25 (a₁) and the 20th term is 500 (a₂₀), we can substitute these values into the formula to obtain two equations:

a₂₀ = a₁ + (20 - 1) * d (Equation 1)
500 = 25 + 19d (Equation 2)

Let's solve these equations to find the common difference (d).

First, simplify Equation 1:
a₂₀ = a₁ + 19d

Substitute the given values into Equation 1:
500 = 25 + 19d

Now, subtract 25 from both sides of the equation:
500 - 25 = 19d

Simplify the left side of the equation:
475 = 19d

Finally, divide both sides by 19 to isolate the common difference:
d = 475 / 19

Therefore, the common difference of the arithmetic progression is d = 25.

To find the common difference (d) of an arithmetic progression (A.P.), you can use the formula:

a + (n - 1)d = l

Where:
a = first term
n = number of terms
d = common difference
l = last term

In this case, the first term (a) is 25 and the 20th term (l) is 500.

a + (n - 1)d = l

25 + (20 - 1)d = 500

25 + 19d = 500

Now, let's solve for d:

19d = 500 - 25

19d = 475

d = 475 / 19

d ≈ 25

Therefore, the common difference (d) of the arithmetic progression is approximately 25.