The common ratio r of a geometric sequence satisfies the quadratic equation 2r squared - 3r -2 = 0. If the sum to infinity of the same sequence is 6, explain why, in this case, r can only take on one value, and hence state the common ratio, r. Also, show that the first term 'a' of this sequence is 9.
Thanks so much xx
The only solutions to 2r^2 - 3r -2 = 0 are
r = 2 and r = -1/2. You can prove that by factoring the equation.
The sequence a + ra + r^2 a + ...will not converge if r = 2. Therefore r must be -1/2.
Use the fact that the sum is 6 to figure out the first term, a
6 = a (1 -1/2 + 1/4 -1/8 + ...)
= a [1/(1 + (1/2))] = a*(2/3)
a = 9
167/3
To find the common ratio (r) in a geometric sequence, we need to solve the quadratic equation that describes the relationship between the terms in the sequence. In this case, the quadratic equation is 2r^2 - 3r - 2 = 0.
Step 1: Solve the quadratic equation.
To solve this equation, we can factor it or use the quadratic formula. Let's use the quadratic formula since factoring might not always be possible.
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying the formula to our quadratic equation, we get:
r = (-(-3) ± √((-3)^2 - 4 * 2 * (-2))) / (2 * 2)
r = (3 ± √(9 + 16)) / 4
r = (3 ± √25) / 4
So, r = (3 ± 5) / 4
This gives two possible values for r:
r1 = (3 + 5) / 4 = 8/4 = 2
r2 = (3 - 5) / 4 = -2/4 = -1/2
Step 2: Determine the valid value for the common ratio.
For a geometric sequence to have a finite sum, the absolute value of the common ratio (|r|) must be less than 1. Since |2| is not less than 1, r = 2 is not a valid value for the common ratio in this case.
Therefore, the valid value for r is r = -1/2 (-0.5).
Step 3: Find the first term (a) of the sequence.
Given that the sum to infinity of the sequence is 6, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r)
Substituting the values into the formula, we have:
6 = a / (1 - (-1/2))
6 = a / (1 + 1/2)
6 = a / (3/2)
6 = (2a) / 3
To solve for 'a', we can cross multiply:
6 * 3 = 2a
18 = 2a
a = 18 / 2
a = 9
Therefore, the first term of the sequence (a) is 9.