The sum of the 2nd and 5th terms of a geometric sequence is 3.5. The sum of the 3rd and 6th is -7. Determine the ratio r of the sequence and the first term a.
ar + ar^4 = 7/2
ar^2 + ar^5 = -7
divide and you have
r(1+r^3) / r^2(1+r^3) = -1/2
1/r = -1/2
r = -2
So, a = 1/4
Check. The sequence is
1/4 -1/2 1 -2 4 -8
-1/2 + 4 = 7/2
1 + (-8) = -7
what are you dividing??
5th term and 6th term
To determine the ratio (r) of the geometric sequence and the first term (a), we can use the formulas for the terms of a geometric sequence.
Let's start by considering the second term (a2) and the fifth term (a5) of the sequence. The general formula for the nth term of a geometric sequence is given by:
an = a * r^(n-1)
where a is the first term, r is the common ratio, and n is the position of the term in the sequence.
Given that the sum of the 2nd and 5th terms is 3.5, we can write the equation:
a2 + a5 = 3.5
Substituting the formula, we have:
a * r + a * r^4 = 3.5
Now let's consider the third term (a3) and the sixth term (a6). Using the same formula, we can write:
a3 + a6 = -7
Substituting the formula, we have:
a * r^2 + a * r^5 = -7
We now have a system of two equations:
a * r + a * r^4 = 3.5 (Equation 1)
a * r^2 + a * r^5 = -7 (Equation 2)
To solve this system of equations, we can use substitution or elimination. Let's use substitution:
From Equation 1, we can express a in terms of r:
a = 3.5 / (r + r^4)
Substituting this value of a into Equation 2, we get:
(3.5 / (r + r^4)) * r^2 + (3.5 / (r + r^4)) * r^5 = -7
Simplifying, we have:
3.5r^2 + 3.5r^5 = -7(r + r^4)
Dividing both sides by 3.5:
r^2 + r^5 = -2(r + r^4)
This equation can be rearranged as:
r^5 + 2r^4 + r^2 + 2r = 0
Now, you can solve this equation to find the value of r using algebraic manipulation or numerical methods such as factoring, the rational root theorem, or graphing on a calculator.