Find the value of the sum -r-1n -s-1n -rs 2r 3s where - rs is zero if r - s - rs is one if r-s

The correct option is A 6(6^n 1)5 Given nbsp; δ _rs = 0 nbsp; nbsp; if r≠ s δ nbsp;_rs=1 if r=s ∴∑_r=1^n ∑_s=1^n δ_rs 2^r 3^ ...

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Change of Variables

Find the valu...

Question

Find the value of the sum $n \sum r = 1$ $n \sum s = 1 δ_{r s} 2^{r} 3^{s}$ where $δ_{r s}$ is zero if $r \neq s$ & $δ_{r s}$ is one if $r = s$

\frac{6 (6^{n} - 1)}{5}

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\frac{6 (6^{n} + 1)}{5}

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\frac{5 (6^{n} + 1)}{6}

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\frac{n (6^{n} - 1)}{6}

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n \sum r = 1 n \sum s = 1 S_{r s} 2^{r} 3^{s}

S_{r s} = 0

r \neq s

S_{r s} = 1

r = s

Δ_{r} = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{r - 1} & \frac{1}{r (r + 1)} & sin n θ x & y & z 2^{n} - 1 & \frac{n}{n + 1} & \frac{sin \frac{n + 1}{2} θ sin \frac{n}{2} θ}{sin \frac{θ}{2}} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

\sum_{r = 1}^{n} Δ_{r}

Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} r & n & 6 r^{2} & {2 n}^{2} & 4 n - 2 r^{3} & {3 n}^{3} & 3 n^{2} - 3 n \end{matrix} ∣ ∣ ∣ ∣ ∣

n - 1 \sum r = 0 Δ_{r}

Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 r - 1 & ^{m} C_{r} & 1 m^{2} - 1 & 2^{m} & m + 1 s i n^{2} (m^{2}) & s i n^{2} (m) & s i n^{2} (m + 1) \end{matrix} ∣ ∣ ∣ ∣ ∣

m \sum r = 0 Δ_{r}

Δ_{r} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2^{n} + 1 & 2^{n + 1} & n (n + 2) 1 & 1 & - 1^{n} C_{r} & 2^{r} & 2 r + 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

n \sum r = 0 Δ_{r}

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