Help me this. The first term of the GP exceeds the second term by 4 and the sum of the second and third terms is 2 by 2/3.Find the first four terms?
Oops. I worked an A.P., not G.P.
a = ar + 4
ar + ar^2 = "2 by 2/3"
Now just solve for a and r and then list the first 4 terms.
a = (a+d)+4
(a+d)+(a+2d) = "2 by 2/3" (whatever that means)
Anyway, now just solve for a and d, and then list the first 4 terms.
2 by 2/3
---> old way of saying 2 divided by 2/3
as in 12 รท 3 or 12 by 3
(don' know why they didn't just say 3)
To solve this problem, we need to use the information given about the first term (a), second term (ar), and third term (ar^2) of a geometric progression (GP). Here's how to proceed:
Step 1: Understand the given information.
The first term exceeds the second term by 4: a = ar + 4.
The sum of the second and third terms is 2 + 2/3: ar + ar^2 = 2 + 2/3.
Step 2: Create a system of equations.
Using the information from Step 1, we have two equations:
a = ar + 4 ---- (Equation 1)
ar + ar^2 = 2 + 2/3 ---- (Equation 2)
Step 3: Solve the system of equations.
First, simplify Equation 2 by multiplying both sides by 3 to eliminate the fraction:
3ar + 3ar^2 = 6 + 2
Combining like terms, we get:
3ar^2 + 3ar - 8 = 0
This equation is in quadratic form. Factorizing or using the quadratic formula, we find that r = 1 and r = -8/3 are the roots.
Substituting r = 1 into Equation 1, we can solve for a:
a = ar + 4
a = 1a + 4
a = 4
Therefore, the common ratio (r) is 1, and the first term (a) is 4.
Step 4: Find the first four terms.
Using the values of a and r, we can find the first four terms of the GP:
First term: a = 4
Second term: ar = 4(1) = 4
Third term: ar^2 = 4(1)^2 = 4
Fourth term: ar^3 = 4(1)^3 = 4
So, the first four terms of the GP are: 4, 4, 4, 4.