Question

If ((3-x) + 6 + (7-5x)) is a geometric series, find two possible values for

a) x
b) the common ratio
c) the sum of the first five terms

Answer 1

When given that ((3-x) + 6 + (7-5x)) forms a geometric series, x can be found to be -1 and 3 (though 3 is invalid as it results in division by zero). The valid common ratio is 1.5 for x = -1, and the sum of the first five terms of the series with x = -1 is 121.

Step-by-step explanation:

Given that ((3-x) + 6 + (7-5x)) forms a geometric series, we can determine the value of x and the common ratio. To be a geometric series, each term must be the previous term multiplied by the common ratio (r).

Let's analyze the given terms:

1. The first term (a1) is (3-x).
2. The second term (a2) is 6.
3. The third term (a3) is (7-5x).

For a sequence to be geometric, a2/ a1 = a3/ a2, which gives us:

6 / (3-x) = (7-5x) / 6

Solving for x, we get:

6(6) = (3-x)(7-5x)

36 = 21 - 8x + 5x2

5x2 - 8x - 15 = 0

Using the quadratic formula, we find the values for x:

x = 3 or x = -1.

Having the value of x, we can now calculate the common ratio for each case:

1. For x = 3, the common ratio r = 6 / (3-3) which is not valid since division by zero is undefined.
2. For x = -1, the common ratio r = 6 / (3-(-1)) = 6 / 4 = 1.5.

Therefore, the only valid solution for x is -1, and the common ratio is 1.5.

To find the sum of the first five terms (S5), we use the formula for the sum of a geometric series:

Sn = a1 * (1 - rn) / (1 - r)

For the first five terms with x = -1:

S5 = 4 * (1 - 1.55) / (1 - 1.5)

S5 = 121

Therefore, two possible values for x are 3 (invalid) and -1, the common ratio is 1.5, and the sum of the first five terms is 121.