Sequences and S...

Sum of the series $$2.3.1+3.4.4+4.5.7+...$$ up to $$n$$ terms is

Let $$\displaystyle \left ( a_{n} \right )n\geq 1$$ be an increasing sequence of positive integers such that 1.$$\displaystyle a_{2n}=a_{n}+n$$ for all $$\displaystyle n\geq 1$$ 2.if $$\displaystyle a_{n}$$ is prime, then n is a prime. Prove that $$\displaystyle a_{n}=n,$$ for all $$\displaystyle n\geq 1.$$

Let $$\displaystyle V_{r}$$ denote the sum of the first r terms of an arithmetic progression (AP) whose first term is r and the common difference is $$\displaystyle (2r-1).$$ Let  $$\displaystyle T_{r}=V_{r+1}-V_{r}-2$$ and  $$\displaystyle Q_{r}=T_{r+1}-T_{r}$$ for $$r=1, 2, ...$$

The sum of n terms of $$1^{2}\, +\, (1^{2}\, +\, 2^{2})\, +\, (1^{2}\, +\, 2^{2}\, +\, 3^{2})\, +\, ....$$.

$$\displaystyle 1^{2}+\left ( 1^{2}+2^{2} \right )+\left (1^{2}+2^{2}+3^{2} \right )+...$$ to $$n$$ terms.

Determine the fourth degree expression in n which is equal to $$\displaystyle \sum_{r=1}^{n}r\left ( r+1 \right )\left ( 2r+3 \right ).$$

Observe the given multiples of 37.
$${37\times3=111}$$
$${37\times 6 =222}$$
$${37\times9=333}$$
$${37\times12=444}$$-------------------------------
Find the product of $${37\times27}$$

Sum to $$20$$ terms of the series $$1.{ 3 }^{ 2 }+2.{ 5 }^{ 2 }+3.{ 7 }^{ 2 }+...$$ is

Find the value of 
$$\displaystyle \left ( 1-\frac{1}{2^{2}} \right )\left ( 1-\frac{1}{3^{3}} \right )\left (1 -\frac{1}{4^{2}} \right )\left ( 1-\frac{1}{5^{2}} \right )........\left ( 1-\frac{1}{9^{2}} \right )\left ( 1-\frac{1}{10^{2}} \right )$$

$$\displaystyle \frac{1^{2}}{1} + \frac{1^{2} + 2^{2}}{1 + 2} + \frac{1^{2} + 2^{2} + 3^{2}}{1 + 2 + 3} + .....$$ upto n terms is