QUESTION 3 The following is an arithmetic sequence: \( \begin{array}{ll}3.1 \quad \text { Calculate the value of } p \text {. } \\ 3.2 \quad \text { Write down the value of: } \\ 3.2 .1 \quad \text { The first term of the sequence } \\ 3.2 .2 \quad \text { The common difference } \\ \text { Explain why none of the numbers in this arithmetic sequence are perfect squares. } \\ 3.3\end{array} \)
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Bonus Knowledge
Arithmetic sequences are interesting because they grow by a constant addition, known as the common difference. For example, if you start with a number and consistently add 2, you could create a sequence like 2, 4, 6, 8, and so on. Historically, the concept of arithmetic sequences dates back to ancient civilizations, who used them in everything from astronomy to commerce! To tackle the problem of why none of the numbers in your arithmetic sequence might be perfect squares, consider the properties of squares: they are always congruent to 0, 1, or 4 (mod 5). If your sequence runs into properties where its terms fall outside this category, you'll see no perfect squares appear. Certain arithmetic sequences can be crafted to skip around these numbers, keeping the squares at bay!
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