# Math Help - term of a series

1. ## term of a series

The rth term of a series is given by

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

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

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

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

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

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

$A(1-\frac{B}{27^n})$, where A and B are constants to be found ,
$
u_r=(\frac{1}{3})^{3r-1}\frac{3}{1}+(\frac{1}{3})^{3r-1}

$

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

$

$
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

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

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

$

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

$

I think now you can continue to