Quick AP term

In the HSC Mathematics Advanced and Extension 1 courses, the Sequences and Series topic includes finding the nth term of an arithmetic sequence using the general formula $a_n = a + (n-1)d$. Mastering this straightforward yet powerful technique will save you from tediously listing out dozens of terms. In this post we’ll apply it to calculate $a_{18}$ for a simple arithmetic sequence.

Problem Statement

Find $a_{18}$ for the arithmetic sequence $2,7,12,\ldots$.

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Hints

Use $a_n=a+(n-1)d$ with $a=2$ and $d=5$.

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Solutions

The nth term of an arithmetic sequence is given by $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. We start by noting the first term is $a = 2$. To find the common difference, subtract the first term from the second: $d = 7 - 2 = 5$. With $a$ and $d$ known, we substitute $n = 18$ into the formula to obtain

$$a_{18}=2+17\cdot 5=87.$$

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Takeaways

• First, determine the common difference $d = T_2 - T_1$.
• Once $a$ and $d$ are known, the formula $a_n = a + (n-1)d$ will give you any term directly.
• Always verify your result by checking it fits the pattern (e.g., repeatedly adding $d$ to the given terms should reach your answer).

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Further Readings

HSC Combinatorics, HSC Mechanics, HSC Vectors, HSC Last Resorts