Arithmetic Series; 51 + 58 + 65 + 72 + ... + 1444

[Monkeyseat]

The arithmetic series

51 + 58 + 65 + 72 + ... + 1444

has 200 terms.

a) Write down the common difference of the series.

d=7

b) Find the 101st term of the series.

101st term = a + (n-1)d
101st term = 51 + (100)*7
101st term = 751

c) Find the sum of the last 100 terms of the series.

Sum of the first 99 terms = n/2(2a + (n-1)d)
Sum of the first 99 terms = 99/2(102 + (98)*7)
Sum of the first 99 terms = 39006

Sum of the first 200 terms = 200/2(102 + (199)*7)
Sum of the first 200 terms = 149500

S200 - S99 = 149500 - 39006
S200 - S99 = 110494

Alternatively, I think you can make the 100th term (744) the first term instead and just do:

Sum of the 'first' 101 terms = 101/2(1488 + (100)*7) = 110494

I just wanted to check whether I have done (c) correctly as I have no answers and don't know whether the alternative method above should be the sum of 101 terms or 100 terms. Could somebody check what I have done?

Thanks very much.

 

[tkhunny]

1) You can prove your theory by the Distributive Property. Simply factor our the first term from all remaining terms ans see what you have left.

2) Are you sure you should have quite at 99? Did your index start at 0?

 

[Monkeyseat]

Thanks for replying.

I don't know what you mean in 1 (do these numbers correspond to a and b or are just general points?) - I haven't learnt about that yet. In 2, again I'm unsure. I'm trying to find the sum of term 100 to term 200 if that's what you mean?

Could you point out where I have gone wrong please? I'm mainly wondering about part (c). I have an exam on Thursday so would like to get this sorted.

Thanks!

 

[tkhunny]

Perhaps I misspoke on the "Distributive Property" That would have been correct for a Geometric Sequence. It's the same idea, but a different function for an Arithmetic Sequence.

Let's do six terms:

51 + 58 + 65 + 72 + 79 + 86 = (51 + 58 + 65) + (72 + 79 + 86)
= (51 + 58 + 65) + ((51+21) + (58+21) + (65+21))
= (51 + 58 + 65) + (51 + 58 + 65 + 3*21)

This should suggest to you that sums of pieces are very much related, with some predictable adjustment. The same formulas work in the same ways.

On the second part, you were asked to add up 100 terms. Are you SURE you have 100 terms in each of your sums?

If you start counting at zero, 0, 1, 2, 3, ..., 99 will be 100 terms.
If you start counting at one, 1, 2, 3, 4,..., 100 will be 100 terms.
Your selection of S99 requires some thought. Did you get 100 in each or 99 in one and 101 in the other?

 

[Monkeyseat]

Perhaps I misspoke on the "Distributive Property" That would have been correct for a Geometric Sequence. It's the same idea, but a different function for an Arithmetic Sequence.

Let's do six terms:

51 + 58 + 65 + 72 + 79 + 86 = (51 + 58 + 65) + (72 + 79 + 86)
= (51 + 58 + 65) + ((51+21) + (58+21) + (65+21))
= (51 + 58 + 65) + (51 + 58 + 65 + 3*21)

This should suggest to you that sums of pieces are very much related, with some predictable adjustment. The same formulas work in the same ways.

On the second part, you were asked to add up 100 terms. Are you SURE you have 100 terms in each of your sums?

If you start counting at zero, 0, 1, 2, 3, ..., 99 will be 100 terms.
If you start counting at one, 1, 2, 3, 4,..., 100 will be 100 terms.
Your selection of S99 requires some thought. Did you get 100 in each or 99 in one and 101 in the other?

I did S99 because S200 - S99 gives me the sum of the last 100 terms doesn't it? Which I'm looking for.

Is my answer incorrect?

Thanks.

 

[stapel]

I did S99 because S200 - S99 gives me the sum of the last 100 terms doesn't it? Which I'm looking for.

As was explained previously, unless you started the index at zero, so the first term was a[sub:2demq21d]0[/sub:2demq21d] rather than a[sub:2demq21d]1[/sub:2demq21d], then a[sub:2demq21d]99[/sub:2demq21d] will not be the hundredth term. And, regardless of the initial index, since 99 + 100 does not equal 200, it is unlikely that the 101 terms between a[sub:2demq21d]99[/sub:2demq21d] and a[sub:2demq21d]200[/sub:2demq21d] will give the desired value.

Is my answer incorrect?

Yes. See above, or here.

Eliz.

 

[Monkeyseat]

Both questions are mine.

Sorry I don't understand what you mean starting the index at 0.

Just briefly, should it be:

Sum of the first 100 terms = 100/2(102 + (99)*7) = 39750
Sum of the first 200 terms = 200/2(102 + (199)*7) = 149500

Sum of the last 100 terms therefore is: 109750?

 

[tkhunny]

You already answered this question.

Sum of the first 99 terms = 99/2(102 + (98)*7)
Sum of the first 200 terms = 200/2(102 + (199)*7)

Why did you not use "Sum of the first 100 terms"? Clearly, this is what is wanted. It appears you have fixed it.

 

[stapel]

Why did you not use "Sum of the first 100 terms"? Clearly, this is what is wanted.

Or, since part (b) asked for the value of a[sub:2tg36wj3]101[/sub:2tg36wj3], the poster could have found the sum of the last hundred terms by setting up a[sub:2tg36wj3]101[/sub:2tg36wj3] through a[sub:2tg36wj3]200[/sub:2tg36wj3] as a new series {b[sub:2tg36wj3]i[/sub:2tg36wj3]}:

Let the old a[sub:2tg36wj3]101[/sub:2tg36wj3] be the new b[sub:2tg36wj3]1[/sub:2tg36wj3] = 751 and let the old a[sub:2tg36wj3]200[/sub:2tg36wj3] be the new b[sub:2tg36wj3]100[/sub:2tg36wj3] = 1444. Then the sum of the last hundred terms (being the value of the sum of the b[sub:2tg36wj3]i[/sub:2tg36wj3]'s) is (100/2)(751 + 1444), and so forth.

Eliz.