Sigma Notation
Neil Trivedi
Teacher
Sigma Notation
Sigma notation, denoted by the Greek letter , is a concise way of representing the sum of a sequence of terms. The general form of sigma notation is:
This means to sum the sequence from its term to its term.
For example:
This means we are summing the sequence from its term to its term.
Example 1:
A sequence is defined deductively by .
a) Write down the first terms of the sequence.
Single Step: Substitute in values of from to into the given formula.
Thus, the first terms are: and .
b) Find the term of the sequence.
Single Step: Substitute into the formula.
c) Find .
Single Step: Sum the first 4 terms of the sequence.
No answer provided.
Forms for Arithmetic and Geometric Sequences
Arithmetic sequences are in the form , where
is the common difference (remember from GCSE that the coefficient of tells you what
the sequences increases by, do not confuse this with representing the first term in
)is the term (when )
is the term position
If we see a summation in the form.
we should recognise this as an arithmetic series.
Geometric sequences are in the form , where is a constant multiplier, is the common ratio, and is the term position. If we see a summation in the form.
we should recognise this as a geometric series.
Example 2:
Find the value of .
Step 1: Identify the type of series, the term, and the common difference/ratio.
This is an arithmetic series, where the term is , and the common difference is .
Step 2: We are summing from the term to the term, so we use the sum formula for an arithmetic series, where and .
No answer provided.
Example 3:
Find the value of .
Step 1: Identify the type of series, the term, and the common difference/ratio.
This is a geometric series, where the term is , and the common ratio is .
Step 2: We are summing from the term to the term, so we use the sum formula for a geometric series, where and .
No answer provided.
In the next example, we’ll look at summing from an value that is not namely, from
to . It is common for students to think that we are summing terms (i.e, ). However, in fact, we are adding terms. This is because both the and terms are included in our calculation.
Example 4:
Find the value of .
Step 1: Identify the type of series, the term, and the common difference/ratio.
Note that we take the “ term” to be the term of the sequence , since we are summing from the term up to and including the term.
This is an arithmetic series, where the “ term” is , and the common difference is .
Step 2: We are summing from the term to the term, so we use the sum formula for an arithmetic series, where and .
No answer provided.
Example 5:
Given that find .
Step 1: Identify the type of series, the term, and the common difference/ratio.
This is a geometric series where the term is and the common ratio is .
Step 2: We are summing from the term to the term, so we use the sum formula for a geometric series where and .
Step 3: Solve for
Multiplying both sides by ,
Dividing both sides by ,
Move the over to the left and the over to the right.
Take on both sides.
No answer provided.
Challenging Question