arithmetic sequences and series
- Thread starter nabb777
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pka
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KNOW! \(\displaystyle \sum\limits_{k = 1}^n k = \frac{{n(n + 1)}}{2}\)
If \(\displaystyle M<N\) then \(\displaystyle \sum\limits_{k = M}^N k = \sum\limits_{k = 1}^N k - \sum\limits_{k = M - 1}^N k \)
HallsofIvy
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Are you saying you do not know what this means? The "..." means "continue in the same way". Since before that each number just increased by 1 that would be 8+ 9+ 10+ 11+ 12+ 13+ 14+ 15. Surely you can add that? 8+ 9= 17. Adding 10, we get 27. adding 11 we get 38. Adding 12 we get 50. Adding 13 we get 63. Adding 14 we get 77. Finally, adding 15, we get 92. 8+ 9+ 10+ ... + 15= 92.
A cute way to do this is to write sum
8 + 9+10+11+12+13+14+15 and then reverse it:
15+14+13+12+11+10+ 9+8
If we add "vertically", 8+ 15= 23, 9+ 14= 23, 10+ 13= 23, etc. Each pair adds to the same thing, 23, because at each step we are adding 1 to the first row but subtracting 1 from the second. So we have 8 pairs of numbers each of which add to 23. The two rows add to 8(23)= 184. Since the two rows are identical, each row adds to 184/2= 92
Another way of saying the same thing is that "the average value of an arithmetic sequence is just the average of the first and last numbers. Here the first and last numbers are 8 and 15 which have average (8+ 15)/2= 23/2. So the average of all eight numbers is 23/2 and their sum is (23/2)(8)= (23)(4)= 92.
But for just a few numbers like this, the simplest thing to do is just what it says: add them!
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