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When we look at a sequence, our first step is to identify any patterns in the arrangement of the numbers. In this example, the sequence is: 1, 4, 9, 16, 25, 36. To see a pattern, one useful approach is to think about the properties of the numbers:
Notice that each term in this sequence is a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself. For instance, 1 is 1 times 1, 4 is 2 times 2, and so on.
Identifying sequences like these helps us understand how they are formed, making it easier to express them in more compact ways, such as using sigma notation.

Perfect squares are key to recognizing patterns in many numerical sequences. In our example:
The number 1 can be written as 1².
The number 4 can be written as 2².
The number 9 can be written as 3².
The number 16 can be written as 4².
The number 25 can be written as 5².
The number 36 can be written as 6².
This shows that the sequence is made up of terms that follow the form i², where i is an integer. Recognizing perfect squares helps you see the structure of sequences and makes them easier to handle mathematically. When you spot a sequence of perfect squares, you can use this knowledge to simplify your work and translate it into a more useful mathematical form.

Summation notation (sigma notation) allows us to write long sums in a compact and readable form.
In summation notation, the Greek letter sigma (Σ) represents the sum. It includes details on what is being summed and over which range.
In our example, we want to sum the squares of the first six positive integers. We identified our pattern as i². We then specify the range of i from 1 to 6:
i = 1: 1²
i = 2: 2²
i = 3: 3²
i = 4: 4²
i = 5: 5²
i = 6: 6²
Using summation notation, this sum is written as:\[ \sum_{i=1}^{6} i^{2} \]This tells us to sum the squares of i, starting from i=1 and ending at i=6. Summation notation makes it straightforward to write and understand sums without having to list out all the terms explicitly.