Sequences and S...

If $$\underset{r = 1}{\overset{n}{\sum}} r (r + 1) (2r + 3) = an^4 + bn^3 + cn^2 + dn + e$$, then 

$$\displaystyle \sum^{n}_{r=0} (-1)^r \,{^nC_r}. \dfrac{(1 + r \ell n 10)}{(1 + \ell n 10^n)^r} =$$

The sum to $$n$$ terms of the services $$\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+.......$$ is -

If $$ (1^2-t_1)+ (2^2-t_2) +.......+(n^2-1)$$, then $$t_n$$ is

The unit's place digit in $$(1446)^{4n + 3}$$ is

The sum of the infinite series, $$ { 1 }^{ 2 }-\dfrac { { 2 }^{ 2 } }{ 5 } +\dfrac { { 3 }^{ 2 } }{ { 5 }^{ 2 } } -\dfrac { { 4 }^{ 2 } }{ { 5 }^{ 3 } } +\dfrac { { 5 }^{ 2 } }{ { 5 }^{ 4 } } -\dfrac { { 6 }^{ 2 } }{ { 5 }^{ 3 } } + ..........  $$ is

The last three digits in $$10!$$ are ?

The sum $$2 \times 5 + 5 \times 9 + 8 \times 13 +  \ldots 10$$ term is