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Summing Sequences

Arithmetic Sequences
1 , 5 , 7 , 12 ,... is an example of a sequence of numbers. The (...) at the end means that the sequence continues indefinitely. Each number in a sequence is called a term. In the sequence shown, 1 is the first term, 12 is the fourth term and we call the general term the nth term.
The sequence1 , 4 , 7 , 10 ,... is a special sequence because the difference between any two consecutive terms is the same. We say that the sequence has a common difference of 3. Such a sequence of numbers which differ from the next number by the same amount is called an arithmetic sequence. We can represent an arithmetic sequence as a + (a+d) + (a+2d) + (a+3d)+...to n terms ( that is there are n terms in the sequence ) where the variable "a" is the first term and "d" is the common difference.

Look closely at the sum : 1+2+3+4+...+17+18+19+20 .
Do you notice that the outside terms (that is term one and term twenty) add to 21.

Now go to the next two terms moving towards the middle terms (terms 2 and 19) - they add to 21 as well.

In fact we can take each pair of numbers in a similar fashion and they will all add to 21. Since we have an even number of terms there will be 10 pairs of numbers all adding to 21.

So 1+2+3+...+20 must add to 10x21=210.

This method of adding is often called Gauss' method and works for any sequence of numbers which have the same common difference between consecutive terms. For example 2 + 5 + 8 + 11 + 14 +...+26 + 29.


Look below for the rule for adding consecutive integers:

The rule

 To find the sum of the first n consecutive numbers (i.e. 1+2+3+...+n) you halve the number n
and times it by one more than n.
The sum of 1+2+3+...+n = n(n+1)/2

There are in fact some general rules which allow you to do some interesting calculations with arithmetic sequences.

General rules

We will use as an example the sum of the sequence 1 + 3 + 5 + ...
For the general sequence :
a + (a+d) + (a+2d) + (a+3d)+...to n terms ( that is there are n terms in the sequence )
1.The first term is a.
e.g. a = 1
2.The common difference is d.
e.g. d = 2
3.The nth term is called tn = a + (n-1)d.
e.g. t4 = 1 + (4-1)2 = 7
4.The sum of the first n terms is Sn= n/2{2a+(n-1)d}.
e.g. S4= 4/2{2.1+(4-1)2} = 16
5. If you know the first term a and the last term tn we can use the rule Sn= n/2{a+ tn}.
e.g. For 1 + 3 + ...+ 7 S4= 4/2{1+ 7} = 16

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