If the sum of \(N\) consecutive odd integers is \(N^2\) , what is the sum of all odd integers between 13 and 39 inclusive?
I calculated this as (the stem says N consecutive odd integers):
\((39-13)/2 + 1=14\)
\(14^2=196\), which is not in answer list.
OE says calculate the range 1-39 and subtract the range 1-11. \(20^2 - 6^2\)
I think this applies if the question stem says "sum of 1st N consecutive odd integers?"
Else please comment on which part is causing the misunderstanding...
Yes. It has to be N consecutive odd integers starting from 1.
In fact, this is a property and not just specific for this particular question.
The property is that for n consecutive odd integers starting from 1 (or -1 when all values are negative),
(a) sum = \(n^2\) for all positive values.
(b) sum = \(-n^2\) for all negative values.
Another interesting property is that for n consecutive even integers starting from 0 (0 included) ,
(a) sum = \(n^2 - n\) for all positive values.
(b) sum = \(-(n^2 - n)\) for all negative values.
Note: In either case, the property is not valid when the set contains both positive and negative values.
Ps. We can even combine both of them to find the sum of N consecutive integers starting from 0.
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