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

Answer 1

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.

Answer 2

are you sure thats correct

Answer 3

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.

Answer 4

show your work

Answer 5

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.