The sum of the first and third terms of a G.P.is 7/3 and the sum of the second and fourth term is 7/4.Fin the first and the common difference
Use your definitions:
a + ar^2 = 7/3
a( 1 + r^2) = 7/3 **
ar + ar^3 = 7/4
a(r + r^3) = 7/4 ***
divide *** by **
(r + r^3)/(1 + r^2) = (7/4) / (7/3) = 3/4
4r + 4r^3 = 3 + 3r^2
4r^3 - 3r^2 + 4r - 3 = 0
r^2(4r - 3) + 1(4r-3) = 0
(4r-3)(r^2 + 1) = 0
4r = 3 or r^2 = -1 , the last is not possible, so
r = 3/4
sub back into ** to find a
btw, in a GP we have a common ratio , not a common difference.
Thanks very helpful
To find the first term and the common difference of a geometric progression (G.P.), you can use the formulas for the terms of a G.P.
Let the first term be 'a' and the common ratio be 'r'.
The terms of the G.P. can be expressed as:
1st term: a
2nd term: ar
3rd term: ar^2
4th term: ar^3
Given that the sum of the first and third terms is 7/3, we have:
a + ar^2 = 7/3 ----(1)
And given that the sum of the second and fourth terms is 7/4, we have:
ar + ar^3 = 7/4 ----(2)
We have two equations (1) and (2) with two variables (a and r). We can solve these equations simultaneously to find the values of 'a' and 'r'.
Let's start with Equation (1):
a + ar^2 = 7/3
Factoring out 'a', we get:
a(1 + r^2) = 7/3
Dividing both sides by (1 + r^2):
a = (7/3) / (1 + r^2)
Now, let's solve Equation (2):
ar + ar^3 = 7/4
Factoring out 'ar', we get:
ar(1 + r^2) = 7/4
Dividing both sides by (1 + r^2):
ar = (7/4) / (1 + r^2)
Using the value of 'a' from Equation (1), we substitute it into Equation (2) to eliminate 'a':
[(7/3) / (1 + r^2)] * r = (7/4) / (1 + r^2)
Now, we can solve this equation for 'r' and then substitute the value of 'r' back into Equation (1) to find 'a'.
Please note that the actual calculation of 'r' and 'a' will require further algebraic manipulation and solving equations.
To find the first term and the common difference of a geometric progression (G.P.), we can use the following formulas:
The formula for the nth term of a G.P. is given by: T(n) = a * r^(n-1)
The formula for the sum of the first n terms of a G.P. is given by: S(n) = a * (1 - r^n) / (1 - r)
Let's solve the problem step by step:
Step 1: Set up the given information
We are given the following:
The sum of the first and third terms of the G.P. is 7/3.
The sum of the second and fourth terms of the G.P. is 7/4.
Step 2: Express the given information using the formulas
Using the formulas, we can express the given information as follows:
(a + ar^2) = 7/3 (1)
(ar + ar^3) = 7/4 (2)
Step 3: Solve the equations
To solve the equations, we can use a method known as substitution. Let's solve equation (2) for 'ar' and substitute it into equation (1):
From equation (2), we can express 'ar' as (7/4 - ar^3).
Substituting this value into equation (1), we get:
(a + (7/4 - ar^3)r^2) = 7/3
Now, simplify the equation:
a + (7/4)r^2 - ar^5 = 7/3
Step 4: Simplify the equation
Multiply through by 12 to eliminate the fractions:
12a + 21r^2 - 12ar^5 = 28
Step 5: Rearrange the equation
Rearrange the equation by grouping common terms:
-12ar^5 + 21r^2 + 12a - 28 = 0
Step 6: Solve for 'ar'
The equation we obtained is a fifth-degree equation in 'ar'. To solve it, we can use numerical methods or approximations.
Unfortunately, as an AI text-based bot, I am unable to solve high-degree equations numerically. However, you can use algebraic calculators or software like Wolfram Alpha to find the solution if needed.
Once you find the value of 'ar', you can substitute it back into either equation (1) or (2) to solve for 'a'.
Please note that this solution method assumes that the given terms in the problem are correct and there are no mistakes or missing information.