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This essential trains for: SAT-II, AMC-12, AIME.

The summation symbol is a cosmetic shorthand for sums that have a large or unspecified number of terms. Instead of writing the sum in expanded form:

we can write:

Notice that j is an integer variable that was not there before. It is called the summation index and it is used to count the terms of the sum.

The lowest value of j is called the lower summation limit and the highest value of j is called the upper summation limit.

The summation symbol is like a machine that works like this:

- set the sum equal to zero
- set the index to the lowest value
- compute the first term
- add the first term to the sum
- REPEAT: increase the index by 1
- compare the index to the upper limit: if smaller or equal, continue the process, if larger the sum is complete and stop
- compute the next term
- add the term to the sum
- go back to REPEAT

The expression of the term can have various forms. It may or may not depend on the summation index.

In the first example above, the term is exactly equal to the summation index.

Example 1:

The term is constant (does not depend on the summation index). The sum is simply:

Example 2:

Another sum with constant term:

Example 3:

In this sum, the term's value alternates between 1 and -1:

If N is even the sum is equal to zero. If N is odd the sum is equal to -1.

Example 4:

This is a geometric sum:

Example 5:

Change the sum below to start counting the terms at 0 instead of 1:

Notice that both limits as well as the expression of the term must be changed:

Now change the same sum to start at 5:

Properties of summation symbols:

Constants can be factored out:

Denoting the general term by:

this property can be written as:

Sums can be re-arranged:

Using both properties we can solve the (arithmetic sequence) example: