For an A.P.

sum of 2n terms = 3* sum of n terms
then
sum of 3n terms : sum of n terms = ?

To find the ratio between the sum of 3n terms and the sum of n terms of an arithmetic progression (A.P.), we first need to determine the common difference (d) of the A.P.

Let's assume the first term of the A.P. is 'a' and the common difference is 'd'. Therefore, the sum of n terms is given by the formula:

Sum of n terms = (n / 2)(2a + (n - 1)d)

Now, let's calculate the sum of 2n terms using the given condition:

Sum of 2n terms = 3 * Sum of n terms
Therefore:
(n / 2)(2a + (2n - 1)d) = 3 * (n / 2)(2a + (n - 1)d)

Since n is a non-zero value, we can cancel out these terms on both sides of the equation:

2a + (2n - 1)d = 3(2a + (n - 1)d)

Now, let's solve this equation to find the value of 'd':

2a + 2nd - d = 6a + 3nd - 3d

Rearranging the equation, we get:

2nd - 3nd = (6a - 2a) + (3d - d)

Simplifying further, we have:

-nd = 4a + 2d

Now, let's express 'a' in terms of 'd':

a = (-nd - 2d) / 4 = (-n - 2)d / 4 = -(n + 2)d / 4

Substituting this value of 'a' in the original equation for the sum of n terms:

Sum of n terms = (n / 2)(2 * (-(n + 2)d / 4) + (n - 1)d)

Simplifying this expression gives us:

Sum of n terms = (n / 2)(-nd - 2d + nd - d)

Notice that the 'nd' terms cancel out:

Sum of n terms = (n / 2)(-2d - d) = (n / 2)(-3d)

Therefore, the sum of n terms is equal to -(3n / 2)d.

Now, let's find the sum of 3n terms. We use the same formula:

Sum of 3n terms = (3n / 2)(2a + (3n - 1)d)

Substitute the expression we found for 'a' earlier:

Sum of 3n terms = (3n / 2)(2 * (-(n + 2)d / 4) + (3n - 1)d)

Simplifying this expression gives us:

Sum of 3n terms = (3n / 2)(-nd - 2d + 3nd - d)

Again, notice that the 'nd' terms cancel out:

Sum of 3n terms = (3n / 2)(-2d - d) = (3n / 2)(-3d)

Therefore, the sum of 3n terms is equal to -(9n / 2)d.

Finally, let's calculate the ratio between the sum of 3n terms and the sum of n terms:

Ratio = Sum of 3n terms / Sum of n terms
Ratio = (-(9n / 2)d) / (-(3n / 2)d)
Ratio = (-9n / 2d) / (-3n / 2d)
Ratio = -9n / -3n
Ratio = 3

Therefore, the ratio between the sum of 3n terms and the sum of n terms of the given A.P. is 3.