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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

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