Question

Hi, could someone please help me with these problems step by step? I am really stuck. These problems are from a "Find the Error" Assignment.

http://assets.openstudy.com/updates/attachments/51945e4ee4b0d270317f72d7-melidalkae-1368678016525-chapter9findtheerrors1.pdf
(They are on the last page and they are the last 3 problems!)

Thank you :)


6
2) Σ n^2+1 What is the sum of the series?
n=1

3) 5
Σ 16(1/4)^n What is the sum of the series?
n=0

4) Does the series 1/4-1/2+1-2+....converge or diverge? If it converges, find the sum.

Answer 1

@whpalmer4

Answer 2

Okay, you aren't supposed to find the answer so much as find the mistake made, as I read it.

Answer 3

For the first one,
\[\sum _{n=1}^6 \left(n^2+1\right)\]

Is that arithmetic, geometric, what?

Answer 4

Well you are supposed to 1) Find the error 2) Correct the error 3) Give the right answer. Ive done that for all of the problems except these 3 :/

Don't laugh at me but I really do not know :/ My math teacher isn't the best at explaining

Answer 5

Can you tell me the defining characteristics of an arithmetic sequence? How about geometric?

Answer 6

An arithmetic goes from one term by either adding or subtracting whereas geometric goes from one term multiplying or dividing, I think?

Answer 7

Okay, that's pretty close. Does \(n^2+1\) fit either of those descriptions?

Answer 8

Let's look at the terms generated:
\[n=1, \,n^2+1 = 1^2+1=2\]\[n=2,\,n^2+1=2^2+1=5\]\[n=3,\,n^2+1=3^2+1=10\]\[n=4,\,n^2+1=4^2+1=17\]
...

Answer 9

With an arithmetic sequence, there is a common difference between terms. Is there one here?

\[5-2=3\]\[10-5=5\]\[17-10=7\]

I think that might be a "no"

Answer 10

With geometric, there is a common ratio between terms. Is there one here?
\[5/2= 2.5\]\[10/5=2\]\[17/10=1.7\]Again, fails the test.

Answer 11

So, this is not an arithmetic or geometric sequence.

Answer 12

Ohh so can the problem still be done?

Answer 13

If you look at the work the idiot who made the mistake did, how did they find the sum?

Answer 14

Aren't they using a formula that applies to arithmetic sequences?

Answer 15

Ohh, so the mistake is that they used an arithmetic formula when it is not either an arithmetic or geometric sequence. For the answer? Is there a way to find the sum if it's neither?

Answer 16

Yes, that's the error they made.

Of course it is possible to find the sum of this itty bitty sequence. One way would be to simply list all 6 terms and add them up!

There is a formula for the sum of squares, which could be adjusted to give you the sum of squares + 1. I suspect you haven't encountered it yet, so that may not be what they are looking to see.

\[\sum _{i=1}^n i^2 = \frac{1}{6} n (1+n) (1+2 n)\]

That means we can take the sum of \(n^2\) by calculating
\[\frac{1}{6}*(6)(1+6)(1+2*6) = \]

and then adding \(6*1\) to the total (to account for the \(+1\) on each term)

Answer 17

My guess is they expect you to just list the terms and add them up, but that's just a guess.

Answer 18

Ohhh I see I would just have to do 1^2+1, 2^2+1, 3^2+1, and so on until I get to 6. I got for the sum 97.

Answer 19

Yes, 97 is correct.

Sorry, I now have to go run those errands I mentioned previously :-(

Answer 20

It's fine! :) Whenever you have time, could you help with the other 2? Thanks for helping as usual :)

Answer 21

@whpalmer4 I know that you're probably out still running errands but I did the second one. Of course when you get the chance to look at it. It is a geometric sequence they were right about that, but for the sum I got 0. Is that possible?

I found their mistake for A(1) they plugged in 16 for that equation but the first term should be 0, right? because it says n=0

Answer 22

The sum of the terms in the second one doesn't equal 0 — aren't they all positive numbers? How do you add a bunch of positive numbers (albeit positive numbers which get smaller and smaller) and end up with 0?

Answer 23

I think I might see what did:
n=0= 16(1/4)^0=16
n=1= 16(1/4)^1= 4
n=2= 16 (1/4)^2=1
n=3= 16 (1/4)^3= .25
n=4= 16 (1/4)^4= .0625
n=5= 16 (1/4)^5= .015625

so is the answer: 21.328125?

Answer 24

It is.

Answer 25

or 1365/64 if you like a nice, neat fraction

Answer 26

Alright thank you, but i am confused what did the original problem do wrong? :/ It's on the last page Number 3.

Answer 27

Look carefully at the starting point and ending point of the sum as they did it, and the sum as you did it. Why don't you observe the totals you get if you add up the numbers one at a time, starting with 16...

Answer 28

I want to say they plugged in the wrong number for the variable 'n', right?

Answer 29

Or is the error was that they weren't supposed to use that formula because it doesn't pertain to this problem?

Answer 30

Nvm I figured it out, the n should be a 6 not a 5 because we found the sum of SIX terms not 5. Thank you @whpalmer4 :) You rock!

Answer 31

You're welcome!