How to Convert a Sequence into Sigma Notation

Turning a sequence into sigma notation can be done by following these simple steps:

1. Identify the Sequence:

   • First, look at the sequence. For example, if you have $a_n = 2n + 3$ for $n = 1, 2, 3, \ldots, N$.

2. Find the Pattern:

   • Try to see if there is a pattern or formula. For instance, if you have the sequence $5, 8, 11, 14, \ldots$, this is called an arithmetic sequence.

3. Write the General Term:

   • Create a formula for the $n$-th term. The sequence $5, 8, 11, 14, \ldots$ can be written as $a_n = 5 + 3(n - 1)$ or $a_n = 3n + 2$.

4. Set the Range for Summation:

   • Identify where the sequence starts and ends. If you start at $n=1$ and go to $N$, your range is from $n=1$ to $n=N$.

5. Write the Sigma Notation:

   • Now, combine the formula you found and the bounds you set. For our earlier sequence, it looks like this: $\sum_{n=1}^{N} (3n + 2)$.

Example

If you want to add the first 5 terms of the sequence $2n + 3$, you do the following:

1. Find the terms: $5, 8, 11, 14, 17$.
2. General term: $a_n = 2n + 3$.
3. Sum in sigma notation: $\sum_{n=1}^{5} (2n + 3)$.

Important Notes

• The index of summation (which is $n$ here) is important. It tells you which term you are calculating.
• Learning to express a series in sigma notation can make it easier to work with complex math later on.
• Sigma notation helps simplify large or complicated sequences.