if the 2nd and 5th term of a G.P is -6 and 48 respectively. find the sum of the first fourth term
3d = a_5 - a_2 = 54
S4 = 4/2 (2a+3d)
To find the sum of the first four terms of a geometric progression (G.P.), we first need to find the common ratio (r) of the progression.
The formula for the nth term of a G.P. is given by:
tn = a * r^(n-1),
where tn is the nth term, a is the first term, r is the common ratio, and n is the number of terms.
We are given the 2nd term as -6, so we can substitute the values into the formula to find a * r. Therefore:
-6 = a * r^(2-1) = a * r.
We are also given the 5th term as 48, so we can substitute the values into the formula:
48 = a * r^(5-1) = a * r^4.
Now, we have a system of two equations with two variables:
-6 = a * r, (Equation 1)
48 = a * r^4. (Equation 2)
To eliminate a, we can divide Equation 2 by Equation 1:
48 / (-6) = (a * r^4) / (a * r).
Simplifying this, we get:
-8 = r^3.
Now, we can find the value of r by taking the cube root of both sides:
r = ∛(-8) = -2.
Substituting this value of r back into Equation 1, we get:
-6 = a * (-2).
Simplifying this, we find:
a = 3.
So, the first term (a) of the G.P. is 3 and the common ratio (r) is -2.
Now, we can find the first four terms of the G.P.:
1st term = a = 3,
2nd term = a * r = 3 * (-2) = -6,
3rd term = a * r^2 = 3 * (-2)^2 = 12,
4th term = a * r^3 = 3 * (-2)^3 = -24.
To find the sum of the first four terms, we add them up:
Sum = 3 + (-6) + 12 + (-24) = -15.
Therefore, the sum of the first four terms of the G.P. is -15.
To find the sum of the first fourth term of the geometric progression (G.P.), we need to first determine the common ratio (r) of the G.P.
Given that the 2nd term is -6 and the 5th term is 48, we can use this information to calculate the common ratio (r).
The formula for the nth term of a G.P. is given by: an = a * r^(n-1), where 'an' represents the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.
Using the given information, we have:
a2 = a * r^(2-1) = -6 (since the 2nd term is -6)
a5 = a * r^(5-1) = 48 (since the 5th term is 48)
Dividing the equation for a2 by the equation for a5, we get:
(-6) / 48 = (a * r) / (a * r^4)
-1/8 = 1 / r^3
(r^-3) / (r^-3) = 1 / r^(-3)
r^(-3) = -8
Taking the reciprocal of both sides of the equation, we get:
r^3 = -1/8
Now, taking the cube root of both sides:
r = -(1/2)
Now that we have found the common ratio (r = -1/2), we can determine the first term (a) using the formula for a2:
a2 = a * r^(2-1) = -6
Substituting the value of r, we can solve for a:
-6 = a * (-1/2)^(2-1)
-6 = a * (-1/2)
-6 = -a/2
Multiplying both sides by -2 to isolate 'a':
12 = a
Hence, the first term (a) of the G.P. is 12, and the common ratio (r) is -1/2.
Now, to find the sum of the first four terms, we'll use the sum formula for a G.P. given by:
S = a * (1 - r^n) / (1 - r), where 'S' represents the sum of the first 'n' terms.
Using the formula, we can determine the sum of the first four terms (n = 4):
S = 12 * (1 - (-1/2)^4) / (1 - (-1/2))
S = 12 * (1 - 1/16) / (1 + 1/2)
S = 12 * (15/16) / (3/2)
S = (12 * 15) / (16 * 3)
S = 180 / 48
S = 15/4
Therefore, the sum of the first four terms of the G.P. is 15/4.