Question: If $< a_{n} >$ is an arithmetic sequence, then \(\Delta = \left| \begin{matrix} a_{m} & a_{n} & a_...

Question

If $< a_{n} >$ is an arithmetic sequence, then $\Delta = \left| \begin{matrix} a_{m} & a_{n} & a_{p} \\ m & n & p \\ 1 & 1 & 1 \end{matrix} \right|$ equals

Answer 1

Solution

Let a be the first term and d the common difference. Then

$a_{r} = a + (r - 1)d$

a + (m - 1)d & a + (n - 1)d & a + (p - 1)d \\ m & n & p \\ 1 & 1 & 1 \end{matrix} \right| = \left| \begin{matrix} a & a & a \\ m & n & p \\ 1 & 1 & 1 \end{matrix} \right| + d\left| \begin{matrix} m - 1 & n - 1 & p - 1 \\ m & n & p \\ 1 & 1 & 1 \end{matrix} \right|$$ $$= a\left| \begin{matrix} 1 & 1 & 1 \\ m & n & p \\ 1 & 1 & 1 \end{matrix} \right| + d\left| \begin{matrix} m & n & p \\ m & n & p \\ 1 & 1 & 1 \end{matrix} \right| = a.0 + d.0 = 0$$

Question: If \(< a_{n} >\) is an arithmetic sequence, then \(\Delta = \left| \begin{matrix} a_{m} & a_{n} & a_...

Solution