Use the geometric sequence of numbers 1, 1/3, 1/9 , 1/27… to find the following:
b) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms? Carry all calculations to 6 decimals on all assignments.
Answer:
Show work in this space.
c) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 12 terms? Carry all calculations to 6 decimals on all assignments.
Answer:
Show work in this space.
d) What observation can make about the successive partial sums of this series? In particular, what number does it appear that the sum will always be smaller than?
Answer:
why are you not using the formula?
r=1/3
I will be happy to critique your thinking.
i'm not using the formula because i just do not understand algebra at all...please help this is my last assignment.
Hmmm.
The write out the first 10 or 12 terms and add them.
However, I suspect your teacher expects you to use the Sum formula, as that is what was taught.
Ok bob...can you check and see if i did it right? thank you.
Use the geometric sequence of numbers 1, 1/3, 1/9 , 1/27… to find the following:
a) What is r, the ratio between 2 consecutive terms?
Answer: r = 1/3
Show work in this space. Each term is the previous term divided by 3 or
Lol that's easy but I'm late to the question on pour pose I've just been too busy
Answer: Yes, you are correct. The common ratio, r, between consecutive terms is indeed 1/3.
To find the sum of the first 10 terms, you can use the formula for the sum of a geometric series:
S = a * (1 - r^n) / (1 - r)
In this formula:
- S represents the sum of the series
- a represents the first term (in this case, a = 1)
- r represents the common ratio (in this case, r = 1/3)
- n represents the number of terms (in this case, n = 10)
Plugging in the values into the formula:
S = 1 * (1 - (1/3)^10) / (1 - 1/3)
Calculating this expression using a calculator or computer program, you will find that the sum of the first 10 terms is approximately 1.0998.
Similarly, to find the sum of the first 12 terms, use the same formula but with n = 12:
S = 1 * (1 - (1/3)^12) / (1 - 1/3)
Calculating this expression, you will find that the sum of the first 12 terms is approximately 1.1247.
Now, for part d) regarding the observations about the successive partial sums of the series, we can see that as we add more terms, the partial sums increase but seem to approach a certain value. In this case, it appears that the sum of the series will always be smaller than the number 1.125 (rounded to three decimal places).
I hope this explanation helps. If you have any further questions, feel free to ask.