Determine whether each infinite geometric series converges or diverges. If the series converges, state the sum. 150+30+6+...

This is a finite geometric series with first term a=150, common ratio r=1/5, and number of terms n=4 (since the fourth term is 6). We can use the formula for the sum of a finite geometric series:

S_n = a(1 - r^n) / (1 - r)

S_4 = 150(1 - (1/5)^4) / (1 - 1/5) = 180

Therefore, the series converges to a sum of 180.

are you sure thats correct

I apologize for the confusion. I made an error in my calculations. Since the common ratio, r, is less than 1 in magnitude, the series is convergent. The sum of an infinite geometric series with first term a and common ratio r, provided that |r| < 1, is:

S = a / (1 - r)

In this case, a = 150 and r = 1/5. Therefore:

S = 150 / (1 - 1/5) = 187.5

So, the infinite geometric series 150+30+6+... converges to a sum of 187.5.

show your work

Yes, here is my work:

We have the infinite geometric series:

150 + 30 + 6 + ...

The first term a = 150 and the common ratio r = 30 / 150 = 6 / 30 = 1 / 5.

For an infinite geometric series with common ratio r, the series is convergent if |r| < 1 and divergent if |r| >= 1.

In this case, we have:

|r| = |1/5| = 1/5 < 1

So, the series is convergent.

The sum of an infinite geometric series with first term a and common ratio r, provided that |r| < 1, is given by:

S = a / (1 - r)

In this case, we have:

S = 150 / (1 - 1/5)

S = 150 / (4/5)

S = 150 * (5/4)

S = 187.5

Therefore, the infinite geometric series 150 + 30 + 6 + ... is convergent and its sum is 187.5.

To determine whether the infinite geometric series converges or diverges, we need to examine the common ratio.

Given the series: 150 + 30 + 6 + ...

To find the common ratio (r), we calculate the ratio between each consecutive term.

r = (30/150) = (6/30) = 1/5

Since the common ratio (r) is between -1 and 1 (in this case, r = 1/5 which is less than 1), the series is convergent.

To find the sum (S) of a convergent geometric series, we can use the formula:

S = a / (1 - r)

where 'a' is the first term and 'r' is the common ratio.

In this case, a = 150 and r = 1/5.

S = 150 / (1 - 1/5)
S = 150 / (4/5)
S = 150 * (5/4)
S = 750 / 4
S = 187.5

Therefore, the infinite geometric series converges to a sum of 187.5.

To determine whether an infinite geometric series converges or diverges, we need to examine the common ratio (r) between terms. In this case, we can see that each term is obtained by multiplying the previous term by a common ratio of 1/5.

An infinite geometric series converges if the absolute value of the common ratio is less than 1 (|r| < 1). In this case, |1/5| = 1/5 and since 1/5 is less than 1, the series converges.

To find the sum of a converging infinite geometric series, we can use the formula: S = a / (1 - r), where 'a' represents the first term and 'r' represents the common ratio.

In this series, the first term (a) is 150, and the common ratio (r) is 1/5. So, the sum (S) can be calculated as follows:

S = 150 / (1 - 1/5)
S = 150 / (4/5)
S = 150 * (5/4)
S = 750 / 4
S = 187.5

Therefore, the infinite geometric series 150 + 30 + 6 + ... converges to a sum of 187.5.