If $D_{r} = \left| \begin{matrix} 2^{r - 1} & 3^{r - 1} & 4^{r - 1} \ x & y & \mathcal{z} \ 2^{n} - 1 & (3^{n} - 1)/2 & (4^{n} - 1)/3 \end{matrix} ight|$, then the value of $\sum_{r = 1}^{n}D_{r}$ is equal to

Question

If $D_{r} = \left| \begin{matrix} 2^{r - 1} & 3^{r - 1} & 4^{r - 1} \ x & y & \mathcal{z} \ 2^{n} - 1 & (3^{n} - 1)/2 & (4^{n} - 1)/3 \end{matrix} ight|$, then the value of $\sum_{r = 1}^{n}D_{r}$ is equal to

Dr. Abubacker · Accepted Answer

(c)
We have, $D_{r} = \left| \begin{matrix}
2^{r - 1} & 3^{r - 1} & 4^{r - 1} \
x & y & \mathcal{z} \
2^{n} - 1 & (3^{n} - 1)/2 & (4^{n} - 1)/3
\end{matrix} ight|$
$\Rightarrow \ \sum_{r = 1}^{n}{D_{r} = \left| \begin{matrix}
\sum_{r = 1}^{n}2^{r - 1} & \sum_{r = 1}^{n}3^{r - 1} & \sum_{r = 1}^{n}4^{r - 1} \
x & y & \mathcal{z} \
2^{n} - 1 & (3^{n} - 1)/2 & (4^{n} - 1)/3
\end{matrix} ight|}$
$\Rightarrow \ \sum_{r = 1}^{n}{D_{r} = \left| \begin{matrix}
2^{n} - 1 & (3^{n} - 1)/2 & (4^{n} - 1)/3 \
x & y & \mathcal{z} \
2^{n} - 1 & (3^{n} - 1)/2 & (4^{n} - 1)/3
\end{matrix} ight|}$
$\sum_{r = 1}^{n}{D_{r} = 0\ \ (\because}$ two rows are same)

Q. If $D_{r} = \left| \begin{matrix} 2^{r - 1} & 3^{r - 1} & 4^{r - 1} \\ x & y & \mathcal{z} \\ 2^{n} - 1 & (3^{n} - 1)/2 & (4^{n} - 1)/3 \end{matrix} \right|$, then the value of $\sum_{r = 1}^{n}D_{r}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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