Sequences and S...

Sum of the series $$S={ 1 }^{ 2 }-{ 2 }^{ 2 }+{ 3 }^{ 2 }-{ 4 }^{ 2 }+......{ 2008 }^{ 2 }+{ 2009 }^{ 2 }$$.

If in a progression $${a_1},{a_2},{a_3},...,etc.,\left( {{a_r} - {a_{r + 1}}} \right)$$ bears a constant ratio with $${a_r}.{a_{r + 1}}$$ then the term of the progression are in 

Suppose n be integer than 1, let $$a_n=\frac 1{\log _n2002}$$. Suppose $$b=a_2+a_3+a_4+a_5$$ and $$c=a_{10}+a_{11}+a_{13}+a_{14}$$, Then (b-c) equals

If $$ (20)^{19}+2(21)(20)^{18}+3(21)^{2}(20)^{17} $$ $$ +\ldots+20(21)^{19}=k(20)^{19} $$ then $$  k  $$ is equal to

If $$\frac { 48 }{ (2)(3) } +\frac { 47 }{ (3)(4) } +\frac { 46 }{ (4)(5) } +........+\frac { 2 }{ (48)(49) } +\frac { 1 }{ (49)(50) } =\frac { 51 }{ 2 } +k(1+\frac { 1 }{ 2 } +\frac { 1 }{ 3 } +.....+\frac { 1 }{ 50 } )$$, then K equals

$${ C }^{ 2n }{ C }_{ n }-{ { C }_{ 1 } }^{ 2n-2 }{ C }_{ n }+{ { C }_{ 2 } }^{ 2n-4 }{ C }_{ n }...$$ equals to 

$${\log _5}2,\,\,{\log _6}\,2,\,\,{\log _{12}}\,\,2\,\,$$ are in 

$$\text{Find the value of   } \log \sin 1^{\circ} . \log \sin 2^{\circ} \ldots \ldots \log \sin 179^{\circ} $$

The value of 1+$${ i }^{ 1 }+{ i }^{ 4 }+{ i }^{ 5 }+.......{ i }^{ 2n }$$ is : 

Find the A.M. of the series $$1,2,4,8,16 , \ldots , 2 ^ { n }$$