(1-1/3)(1-1/4)(1-1/5)_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (1-1/n)

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Let ${a_{n}}$ be a non - constant arithmetic progression with $a_{1} = 1$ and for any $n \geq 1$ , the value $\frac{a_{2 n} + a_{2 n - 1} + \dots + a_{n + 1}}{a_{n} + a_{n - 1} + \dots a_{1}}$ remains constant.

Then, $a_{15}$ will be?

Q. Prove that :

|\begin{matrix} (a + 1) (a + 2) & a + 2 & 1 \\ (a + 2) (a + 3) & a + 3 & 1 \\ (a + 3) (a + 4) & a + 4 & 1 \end{matrix}| = - 2

Q. Matrix

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 1 & 1 & 5 2 & 4 & 7 \end{matrix} ⎤ ⎥ ⎦

then the value of

a_{31} A_{31} + a_{32} A_{32} + a_{33} A_{33}

Q. The value of

(1 + \frac{1}{a}) (1 + \frac{1}{a - 1}) (1 + \frac{1}{a - 2}) (1 + \frac{1}{a - 3}) (1 + \frac{1}{a - 4})

Q. If

A = \sum_{r = 0}^{n} (^{n} C_{r} c o s (n - r) x . s i n r x), B = 2^{n} s i n n x

and

C = \frac{1}{! 0! n - 1} + \frac{1}{! 3! n - 3} + \frac{1}{! 5! n - 5} + \dots + \frac{1}{! n - 1! 1}

, then

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