If Dr-2r-1 23r-1 45r-1 x y z 2n-1 3n-1 5n-1 then prove that -r-1nDr-0

∑_r=1^nD_r=∑_r=1^n2^r 1 amp; ∑_r=1^n2(3^r 1) amp; ∑_r=1^n4(5^r 1) x amp; y amp; z 2^n 1 amp; 3^n 1 amp; 5^n 1 =1+2+...2^n 1 ...

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Byju's Answer

Standard XII

Mathematics

Determinant

If Dr=2r-1 ...

Question

If $D_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{r - 1} & 2 (3^{r - 1}) & 4 (5^{r - 1}) x & y & z 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ then prove that $n \sum r = 1 D_{r} = 0$

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Solution

$n \sum r = 1 D_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} \sum_{r = 1}^{n} 2^{r - 1} & \sum_{r = 1}^{n} 2 (3^{r - 1}) & \sum_{r = 1}^{n} 4 (5^{r - 1}) x & y & z 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 + 2 + . . {.2}^{n - 1} & 2 {1 + {3....3}^{n - 1}} & 4 (1 + 5 + . . . {.5}^{n - 1}) x & y & z 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{2^{n} - 1}{2 - 1} & \frac{2 (3^{n} - 1)}{3 - 1} & \frac{4 (5^{n} - 1)}{5 - 1} x & y & z 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{n - 1} & 3^{n - 1} & 5^{n - 1} x & y & z 2^{n - 1} & 3^{n - 1} & 5^{n - 1} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
Since $R_{1} = R_{3}$

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Similar questions

D_{r}

∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{r - 1} & 2 (3^{r - 1}) & 4 (5^{r - 1}) x & y & z 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

\Rightarrow \sum_{r = 1}^{n} D_{r} =

Q. If

D_{r} =

∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{r - 1} & 2 (3^{r - 1}) & 4 (5^{r - 1}) x & y & 3 z 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

then prove that

n \sum r = 1 D_{r} = 0.

Q. Let

D_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{r - 1} & {2.3}^{r - 1} & {4.5}^{r - 1} α & β & γ 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

. Then, the value of

\sum_{r = 1}^{n} D_{r}

$If D_{r} = ⎛ ⎜ ⎝ \begin{matrix} 2^{r - 1} & 2 . 3^{r - 1} & 4 . 5^{r - 1} α & β & γ 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣, then the value of \sum_{r = 1}^{n} D_{r} is$

Q. If

D_{r} =

∣ ∣ ∣ ∣ ∣ \begin{matrix} r - 1 & n & 6 {(r - 1)}^{2} & 2 n^{2} & 4 n - 2 {(r - 1)}^{3} & 3 n^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣

then

n \sum r = 1 D_{r} =

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If Dr=∣∣ ∣ ∣∣2r−12(3r−1)4(5r−1)xyz2n−13n−15n−1∣∣ ∣ ∣∣ then prove that n∑r=1Dr=0

If $D_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{r - 1} & 2 (3^{r - 1}) & 4 (5^{r - 1}) x & y & z 2^{n} - 1 & 3^{n} - 1 & 5^{n} - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$ then prove that $n \sum r = 1 D_{r} = 0$