### Sequences and Series

goals
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 $\displaystyle x^{2}-36x+\alpha _{2}=0$.