Chrissie B.

asked • 02/16/18

The first 100 counting numbers are arrangedin columns as shown. What would be the sum of all numbers in column c?

Set up like this
 
A     B     C     D     E
1     2     3     4     5
10   9     8     7     6
11   12   13   14   15
20   19   18   17   16
 
we have answer of 968 but we wrote it all out the long way, is there a formula you could do we know the pattern in the column is add 5 each time?

2 Answers By Expert Tutors

By:

Scott S. answered • 02/16/18

Tutor
5.0 (311)

No Stress Math Tutor

Mark M.

The formula that you use, although it produces the correct sum, is for a series that has a common difference (arithmetic).
The numbers under C are not, as listed, do not have a common difference.
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02/16/18

Scott S.

tutor
Am I seeing the columns incorrectly? Sometimes things don't always render accurately on this site, but it looks like the numbers are 3, 8, 13, 18... which would be a common difference of 5
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02/16/18

Chrissie B.

Why the divide by 2?
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02/25/18

Scott S.

tutor
Chrissie, the short answer is that, that is just the formula and you just need to be able to use it as is. If you are looking for a more involved answer...
 
We can write the sum Sn in two different ways.
 
Sn = a1 + (a1 +d) + (a1 + 2d) + . . . + an
Sn = an + (an - d) + (an - 2d) + . . . + a1
 
Adding these two equations together would give you
 
2Sn = (a1 + an) + (a1 + an) + . . . + (a1 + an)    or    2Sn = n(a1 + an)
 
Dividing both side by 2 gives the formula we used above.
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02/28/18

Scott S.

tutor
Chrissie, the short answer is that, that is just the formula and you just need to be able to use it as is. If you are looking for a more involved answer...
 
We can write the sum Sn in two different ways.
 
Sn = a1 + (a1 +d) + (a1 + 2d) + . . . + an
Sn = an + (an - d) + (an - 2d) + . . . + a1
 
Adding these two equations together would give you
 
2Sn = (a1 + an) + (a1 + an) + . . . + (a1 + an)    or    2Sn = n(a1 + an)
 
Dividing both side by 2 gives the formula we used above.
Report

02/28/18

Mark M. answered • 02/16/18

Tutor
5.0 (243)

Mathematics Teacher - NCLB Highly Qualified

Mark M.

I erroneously took the numbers from D rather than C.
The two patterns are 
3, 13, 23, .... and 8, 18, 28, ...
Using the same fomula
The results is still 1010
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02/16/18

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