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# arithmetic sequence

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The fifth term of an arithmetic sequence is 9 and the 32nd term is $$60$$. What is the 23rd term?

May 3, 2022

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Let $$a_n$$ denote the $$n$$th term of this arithmetic sequence. There are 27 terms occurring from $$a_5 = 9$$ to $$a_{32} = 60$$. Assuming a constant common difference between terms, then the terms increase by $$\frac{60-9}{27} = \frac{17}{9}$$ with each $$n$$ step. That is, $$a_{n+1} = a_n + \frac{17}{9}$$.

There are 18 terms occurring from $$a_5 = 9$$ to $$a_{23}.$$ Then, there would be 18 additions of $$\frac{17}{9}$$ from $$a_5$$ to $$a_{23}$$. That is,

$$a_{23} = a_5 + 18\left(\frac{17}{9}\right) = 9 + 34 = 43$$.

May 3, 2022