The third term of a G. P. is 12 and 6th term is 96, then find the sum of nine terms of the G. P.

3rd term is ar^2=12 and 6th term is ar^5=96 To find r we divide 6th term by 3rd term. So r =2. Then substitute in 3rd term and so the 1st term(a) =3. So GP sum of 9 terms is S9=a(r^n-1)/r-1. Substituting we have S9=3(2^9-1)/2-1. So the answer is 1533

To find the sum of nine terms of a geometric progression (G.P), we need to know the first term and the common ratio of the G.P.

Given:
The third term is 12, so we can say that a + 2d = 12. (where a represents the first term and d represents the common difference).

The sixth term is 96, so we can say that a + 5d = 96.

To find the values of a and d, we can solve the system of equations formed by these equations.

Subtracting the first equation from the second equation, we get:
(a + 5d) - (a + 2d) = 96 - 12
3d = 84
Simplifying, we get:
d = 28

Substituting the value of d in the first equation, we get:
a + 2(28) = 12
a + 56 = 12
a = -44

Now that we know the values of a and d, we can find the sum of nine terms of the G.P using the formula:

Sn = (a * (r^n - 1)) / (r - 1)

Where Sn is the sum of n terms, a is the first term, r is the common ratio, and n is the number of terms.

Substituting the values, we get:
Sn = (-44 * ((28^9) - 1)) / (28-1)

Simplifying, we get:
Sn = (-44 * (3869835264 - 1)) / 27

Sn = (-44 * 3869835263) / 27

Sn ≈ -6362651459

Therefore, the sum of nine terms of the G.P is approximately -6362651459.

To find the sum of nine terms of a Geometric Progression (G.P.), we need to first find the common ratio (r) and the first term (a).

Given:
Third term (T₃) = 12
Sixth term (T₆) = 96

We can use the formula to find the nth term of a G.P.:
Tₙ = a * r^(n-1)

Let's solve for the common ratio (r) first:

T₃ = a * r^(3-1)
12 = a * r^2

T₆ = a * r^(6-1)
96 = a * r^5

Divide the second equation by the first equation:

96/12 = (a * r^5) / (a * r^2)

8 = r^3

Taking the cube root of both sides:

r = ∛8 = 2

Now we can find the first term (a) using the third term (T₃):

12 = a * 2^2
12 = 4a

Divide both sides by 4:

a = 3

Now that we have the common ratio (r = 2) and the first term (a = 3), we can find the sum of nine terms (S₉) using the formula for the sum of n terms of a G.P.:

Sₙ = (a * (r^n - 1)) / (r - 1)

Substituting the values:

S₉ = (3 * (2^9 - 1)) / (2 - 1)

Simplify:

S₉ = (3 * (512 - 1)) / 1

S₉ = (3 * 511)

S₉ = 1533

Therefore, the sum of nine terms of the G.P. is 1533.