In a gp the 3rd term is 12.and 6th term is 96.then find the sum of nine numbers.
The sum of first three terms of a G.P. is 13/12 and their product is -1, find the common ratio and the
To find the sum of the first nine terms of a geometric progression (GP), we need to determine the common ratio (r) and the first term (a).
We are given that the 3rd term is 12 and the 6th term is 96. Let's use this information to solve for a and r.
The formula to find the nth term of a GP is:
tn = a * r^(n-1)
Using this formula, we can set up two equations:
For the 3rd term: 12 = a * r^(3-1) = a * r^2 ---- (Equation 1)
For the 6th term: 96 = a * r^(6-1) = a * r^5 ---- (Equation 2)
We can now solve these two equations simultaneously to find the values of a and r.
Dividing Equation 2 by Equation 1, we get:
96/12 = (a * r^5) / (a * r^2)
8 = r^3
Taking the cube root of both sides, we get:
r = 2
Now, substitute the value of r = 2 back into Equation 1 to solve for a:
12 = a * 2^2
12 = 4a
Divide both sides by 4:
a = 3
Therefore, the first term (a) is 3 and the common ratio (r) is 2.
Now, we can calculate the sum of the first nine terms using the formula for the sum of a geometric series:
Sn = a * (1 - r^n) / (1 - r)
Substituting the values:
Sn = 3 * (1 - 2^9) / (1 - 2)
Simplifying:
Sn = 3 * (1 - 512) / (-1)
= 3 * (-511) / (-1)
= 1533
Thus, the sum of the first nine terms of the GP is 1533.
To find the sum of nine numbers in a geometric progression (GP), we first need to find the common ratio (r) of the sequence.
Given that the 3rd term is 12, we can write the equation for the terms of the GP as:
a + 2d = 12 --------(1)
Where "a" is the first term and "d" is the common difference.
Similarly, for the 6th term being 96, we can write:
a + 5d = 96 --------(2)
Now we have two equations with two unknowns. We can solve these equations simultaneously to find the values of "a" and "d".
Subtracting equation (1) from equation (2), we get:
(a + 5d) - (a + 2d) = 96 - 12
3d = 84
d = 84/3
d = 28
Now we can substitute the value of "d" back into equation (1):
a + 2(28) = 12
a + 56 = 12
a = 12 - 56
a = -44
So, the first term "a" is equal to -44, and the common difference "d" is 28.
Now we can find the sum of nine numbers (S9) by using the formula for the sum of a GP:
S9 = a*(1 - r^9) / (1 - r)
Substituting the values, we get:
S9 = (-44)*(1 - 28^9) / (1 - 28)
Simplifying the equation, we calculate:
S9 = (-44)*(1 - 256^9) / (1 - 28)
S9 = (-44)*(1 - 281,474,976,710,656) / (1 - 28)
S9 = (-44)*(1 - 281,474,976,710,656) / (-27)
S9 = 44 * 281,474,976,710,657 / 27
Now, you can divide 44 multiplied by 281,474,976,710,657 by 27 to find the sum of the nine numbers in the geometric progression.
well, r^3 = 96/12 = 8, so r=2
take it from there.