Sequences and Series

Sum the series :
1.1 + 3.01 + 5.001 + 7.0001 ..... n terms.
Which term of the AP : 3, 15, 27, 39,.... will be 132 more than its 54th term?
Prove that $$\displaystyle\sum^n_{k=1}k 2^{-k}=2[1-2^{-n}-n.2^{-(n+1)}]$$.
Given the sequence of numbers $$x_1, x_2, x_3, x_4,....x_{2005}$$ which satisfy $$\dfrac{x_1}{x_1+1}=\dfrac{x_2}{x_2+3}=\dfrac{x_3}{x_3+5}=.....=\dfrac{x_{1005}}{x_{1005}+2009}$$. Also $$x_1+x_2+x_3+......+x_{1005}=2010$$.
Let $$a$$ & $$b$$ are roots of the equation $$\displaystyle x^{2}-4x+\alpha _{1}=0$$ and $$c$$ & $$d$$ are roots of the equation $$\displaystyle x^{2}-36x+\alpha _{2}=0$$.