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# Arithmetic sequence

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Let a_1, a_2, a_3, ... be an arithmetic sequence.

If a_{23} = 2/3 and a_{53} = $$5/3$$, what is a_{35}?

May 14, 2022

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Let d be the common difference.

$$\begin{cases}a_{23} = \dfrac23\\a_{53} = \dfrac53\end{cases} \implies \begin{cases}a_1 + 22d = \dfrac23 \\a_1 + 52d = \dfrac53\end{cases}$$

Solving the linear system for a_1 and d gives a_1 = -1/15, d = 1/30.

Therefore, $$a_{35} = a_1 + 34d = -\dfrac1{15} + \dfrac{34}{30} = \dfrac{16}{15}$$

May 14, 2022