Sequences and S...

$$40280625, 732375, 16275, 465, 18.6, 1.24,?$$

$$435, 354, 282, 219, 165, ?$$

For a positive integer n, let $$f_n(\theta)=(2\cos \theta+1)(2\cos \theta -1)(2\cos 2\theta -1)(2\cos (2^2)\theta -1)......(2\cos (2^{n-1})\theta -1)$$. Which one of the following hold(s) not good?

If $$x=\sqrt 1+\dfrac{1}{{1}^{2}}+\dfrac{1}{{2}^{2}}+\sqrt 1+\dfrac{1}{{2}^{2}}+\dfrac{1}{{3}^{2}}+\sqrt 1+\dfrac{1}{{2019}^{2}}+\dfrac{1}{{2020}^{2}}$$ then $$4040(2020-x)$$

If $$a_{k}=\dfrac{1}{K(K+1)}$$ for $$K=1, 2, 3,.....n,$$ then $$\left(\sum_{k=1}^{n}{a_{k}}\right)^{2}=$$

If $${ x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 },......,{ x }_{ n }\quad $$ are the roots of $$\quad { x }^{ n }+ax+b=0$$. then the value of $$\quad \left( { x }_{ 1 }-{ x }_{ 2 } \right) \left( { x }_{ 1 }-{ x }_{ 3 } \right) \left( { x }_{ 1 }-{ x }_{ 4 } \right) ......\left( { x }_{ 1 }-{ x }_{ n } \right) $$

$$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +2\dfrac { { C }_{ 2 } }{ { C }_{ 1 } } +3\frac { { C }_{ 3 } }{ { C }_{ 2 } } +......+\dfrac { n{ C }_{ n } }{ { C }_{ n-1 } } $$ equals 

The value of $$2+\dfrac{1}{2+\dfrac{1}{2+.....\infty}}$$ is :

If $$\left( 1+x+{ x }^{ 2 } \right) ={ a }_{ 0 }+{ a }_{ 1 }x+{ a }_{ 2 }{ x }^{ 2 }+........{ a }_{ 2n }{ x }^{ 2n }$$, then the value of $${ a }_{ 0 }+{ a }_{ 3 }+{ a }_{ 6 }+.....$$ is 

The sum of the infinite series $$\cot { ^{ 1 } } \left( \dfrac { 7 }{ 4 } \right) +\cot { ^{ -1 }\left( \dfrac { 19 }{ 4 } \right) } +\cot { ^{ -1 }\left( \dfrac { 39 }{ 4 } \right) } +\cot { ^{ -1 }\left( \dfrac { 67 }{ 4 } \right) } +........\infty $$ is: