http://www.ck12.org Chapter 12. Discrete Math
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 3 + 7 + 11 + 15 + 19
= 10 + 26 + 19
= 36 + 19
= 55
Another way is to note that 1+ 10 = 2 + 9 = 3 + 8 = 4 + 7 = 5 + 6 =11. There are 5 pairs of 11 which total 55.
Example B
Evaluate the following sum.
5
k∑= 05 k−^2
Solution:The first term is -2, the last term is 23 and there are 6 terms making 3 pairs. A common mistake is to
forget to count the 0 index.
5
k∑= 05 k−^2 =^62 ·(−^2 +^23 ) =^3 ·^21 =^63
Example C
Try to evaluate the sum of the following geometric series using the same technique as you would for an arithmetic
series.
(^18) + (^12) + 2 + 8 + 32
Solution:
The real sum is:^3418
When you try to use the technique used for arithmetic sequences you get: 3(^18 + 32 )=^7718
It is important to know that geometric series have their own method for summing. The method learned in this concept
only works for arithmetic series.
Concept Problem Revisited
Gauss was a mathematician who lived hundreds of years ago and there is an anecdote told about him when he was a
young boy in school. When misbehaving, his teacher asked him to add up all the numbers between 1 and 100 and
he stated 5050 within a few seconds.
You should notice that 1+ 100 = 2 + 99 =···=101 and that there are exactly 50 pairs that sum to be 101. 50· 101 =
5050.
Vocabulary
Anarithmetic seriesis a sum of numbers whose consecutive terms form an arithmetic sequence.
Guided Practice
- Sum the first 15 terms of the following arithmetic sequence.
− 1 ,^23 ,^73 , 4 ,^173 ... - Sum the first 100 terms of the following arithmetic sequence.
− 7 ,− 4 ,− 1 , 2 , 5 , 8 ,...