The first two terms of geometry series are x + 4 and x²- 16

Question

For which values of x will the series converge
Calculate the sum to infinity of the service in terms of x
If the sum to infinity of the series is 5,determine the value of x

Answer 1

To determine the values of x for which the series will converge, we need to find the common ratio of the geometry series.

The common ratio (r) of a geometry series can be found by dividing any term by its preceding term.

So, (x²- 16) / (x + 4) = r

We can simplify this equation by factoring x² - 16.
(x + 4)(x - 4) / (x + 4) = r

Canceling out the common factor of (x + 4) on both sides, we get:
x - 4 = r

For the series to converge, the absolute value of the common ratio (|r|) must be less than 1.

|x - 4| < 1

This inequality can be solved as:
-1 < x - 4 < 1

Adding 4 to all sides, we get:
3 < x < 5

Therefore, the values of x for which the series will converge are x is greater than 3 and less than 5.

To calculate the sum to infinity of the series, we can use the formula:
S = a / (1 - r)

In this case, the first term (a) of the series is x + 4 and the common ratio (r) is x - 4.

So, S = (x + 4) / (1 - (x - 4))

Simplifying this expression, we get:
S = (x + 4) / (5 - x)

If the sum to infinity of the series is 5, then we can set S equal to 5 and solve for x.

5 = (x + 4) / (5 - x)

Multiplying both sides by (5 - x), we get:
25 - 5x = x + 4

Combining like terms, we have:
6x = 21

Dividing both sides by 6, we get:
x = 3.5

Therefore, the value of x for which the sum to infinity of the series is 5 is x = 3.5.