term of a series

• Feb 25th 2009, 11:07 PM
thereddevils
term of a series
The rth term of a series is given by

$\displaystyle u_r=(\frac{1}{3})^{3r-2}+(\frac{1}{3})^{3r-1}$

Express $\displaystyle \sum^{n}_{r=1}u_r$ in the form

$\displaystyle A(1-\frac{B}{27^n})$, where A and B are constants to be found ,
• Feb 26th 2009, 01:14 AM
Quote:

Originally Posted by thereddevils
The rth term of a series is given by

$\displaystyle u_r=(\frac{1}{3})^{3r-2}+(\frac{1}{3})^{3r-1}$

Express $\displaystyle \sum^{n}_{r=1}u_r$ in the form

$\displaystyle A(1-\frac{B}{27^n})$, where A and B are constants to be found ,

$\displaystyle u_r=(\frac{1}{3})^{3r-1}\frac{3}{1}+(\frac{1}{3})^{3r-1}$

$\displaystyle u_r=(\frac{1}{3})^{3r-1}(3+1)$

$\displaystyle u_r=(\frac{1}{3})^{3r-1}(4) ~= 12\frac{1}{3^{3r}} ~= \frac{12}{27^r}$

No need to tell you that this is a GP

$\displaystyle Sum = a\frac{(r^n-1)}{r-1}$

$\displaystyle =\frac{4}{9} \frac{(1-\frac{1}{27^n})}{(1-\frac{1}{27})}$

$\displaystyle =\frac{4 \times 27}{9\times 26} {(1-\frac{1}{27^n})}$

I think now you can continue to (Dance)