Prove that 1- 2- 3- 2- 3- 4- 3- 4- 5-4-

1! amp; 2! amp; 3! 2! amp; 3! amp; 4! 3! amp; 4! amp; 5! = 1 amp; 2 amp; 6 2 amp; 6 amp; 24 6 am ...

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Properties of Determinants

Prove that ...

Question

Prove that $∣ ∣ ∣ ∣ \begin{matrix} 1! & 2! & 3! 2! & 3! & 4! 3! & 4! & 5! \end{matrix} ∣ ∣ ∣ ∣ = 4!$

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\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{5}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}}

= 2

\frac{2}{1!} + \frac{4}{3!} + \frac{6}{5!} + . . . . . + \infty = e

A = ⎛ ⎜ ⎝ \begin{matrix} 2 & 3 & 4 0 & - 2 & 1 3 & - 1 & 2 \end{matrix} ⎞ ⎟ ⎠, B = ⎛ ⎜ ⎝ \begin{matrix} 2 & 0 & - 3 4 & 0 & - 1 3 & 4 & 5 \end{matrix} ⎞ ⎟ ⎠

C = ⎛ ⎜ ⎝ \begin{matrix} 5 & 6 & 7 - 1 & 2 & 3 4 & - 5 & 4 \end{matrix} ⎞ ⎟ ⎠

A (B C)

(A B) C

\frac{4}{1!} + \frac{11}{2!} + \frac{22}{3!} + \frac{37}{4!} + \frac{56}{5!} + . . . \infty = 6 e - 1

Find the mode from the given data.

1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2 , 3, 4, 1, 2, 3, 4, 1 ,2 , 3, 4, 5, 5, 1, 5, 5, 5, 3, 5, 1, 2, 4, 5, 6, 1, 2, 4, 2, 1

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