Sequences and S...

Let three matrices are $$A=\left[ \begin{matrix} 2 & 1 \\ 4 & 1 \end{matrix} \right] ;A=\left[ \begin{matrix} 2 & 4 \\ 2 & 3 \end{matrix} \right] $$ and $$C=\left[ \begin{matrix} 3 & -4 \\ -2 & 3 \end{matrix} \right] $$, then $${ t }_{ r }(A)+{ t }_{ r }\left( \frac { ABC }{ 2 }  \right) +{ t }_{ r }\left( \frac { A{ (BC) }^{ 2 } }{ 4 }  \right) +{ t }_{ r }\left( \frac { A{ (BC) }^{ 3 } }{ 8 }  \right) +.......+\infty $$ is equal to -

The average expenditure of Sharma for the January to June is Rs. 4200 and he spent Rs. 1200 in January and  Rs.1500 in July. The average expenditure for the months of February to July is:

The value of $$\cot { \left( \sum _{ n=1 }^{ 23 }{ \cot { ^{ -1 } }  } \left( 1+\sum _{ k=1 }^{ n }{ 2k }  \right)  \right)  } $$ is 

The sum of the series (1 + 2)+(1+2+$${ 2 }^{ 2 }$$)+$$(1+2+{ 2 }^{ 2 }+{ 2 }^{ 3 })+.....$$ up to n terms is 

There are n A.M.'s between 3 and 29 such that $$6^{th} mean:(n-1)^{th} mean = 3:5$$, then the value of n, is

If first and $$( 2 n - 1 )^{th}$$ terms of an $$A . P , G . P .$$ and $$H . P$$ . are equal and their $$n th$$ terms are $$a , b , c$$ respectively , then

The $$20^{th}$$ term of  the series $$2\frac{1}{2}+1\frac{7}{13}+1\frac{1}{9}+\frac{20}{23}+......$$  is 

21,25,52,68,193,?

49,64,?,100,121

1001,1004,1012,1027,?