Question

Find the sum of the series or state that the sum does not exist.

5 + 7.5 + 11.25 + ...

and

16 + 12 + 9 + ...

Answer 1

the 16+12+9 doesn't exist right?

Answer 2

Do you know what kind of sequence you're dealing with here?

Answer 3

geometric right?

Answer 4

you are correct

Answer 5

Notice how 7.5/5 = 1.5 and 11.25/7.5 = 1.5, so this shows us that we're multiplying each term by 1.5 to get the next term

Answer 6

yes i see that

Answer 7

Now do you know the rule for summing terms in an geometric sequence? Specifically the rule of summing an infinite number of terms.

Answer 8

r<1?

Answer 9

Close, it's more like |r| < 1

If that is true, then the infinite sum exists.

Answer 10

so this sum does not exist

Answer 11

So in the case of r = 1.5, |r| < 1 is false since |1.5| < 1 is false

Answer 12

but the other one is 64 right?

Answer 13

So you are correct, the infinite sum doesn't exist for the first sequence.

Answer 14

The first term is 16. So a = 16

What is the value of r in the second sequence?

Answer 15

.75

Answer 16

Good, so the infinite sum is

S = a/(1-r)

S = 16/(1-0.75)

S = 64

So you are correct.

Answer 17

thanks!

Answer 18

you're welcome