Question

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

Answer 1

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

Answer 2

167/3