First slide

Introduction to Determinants

Question

Let $Δ_{r} = |\begin{array}{ccc} 2^{r - 1} & 2 (3^{r - 1}) & 4 (5^{r - 1}) \\ α & β & γ \\ 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{array}|$ , for $r = 1, 2, \dots, n ._{r}$ .Then $\sum_{r = 1}^{n} Δ_{r}$

Moderate

A

independent of $α$ , $β$ , $γ$ and $n$
B

independent of $n$ only
C

depends on $α$ , $β$ , $γ$ and $n$
D

independent of $α, β, γ$ only

Solution

$\sum_{r = 1}^{n} Δ_{r} = |\begin{array}{ccc} a & b & c \\ α & β & γ \\ 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{array}|$
where $a = \sum_{r = 1}^{n} 2^{r - 1} = 2^{n} - 1$
$b = \sum_{r = 1}^{n} 2 (3^{r - 1}) = \frac{2 (3^{n} - 1)}{3 - 1}$
$= 3^{n} - 1$
and $\begin{array}{l} c = \sum_{r = 1}^{n} 4 (5^{r - 1}) = \frac{4 (5^{n} - 1)}{5 - 1} \\ = 5^{n} - 1 \end{array}$
Thus, $\sum_{r = 1}^{n} Δ_{r} = |\begin{array}{ccc} a & b & c \\ α & β & γ \\ a & b & c \end{array}| = 0$

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