Hello bhitroofen01 Originally Posted by
bhitroofen01 Hi,
$\displaystyle U_n=\sum_{i=0}^n \frac{1}{3^k}$
$\displaystyle V_n=\sum_{i=0}^n \frac{k}{3^k}$
How to show that: $\displaystyle 3V_{n+1}=V_n+U_n$????
First, note that, when $\displaystyle k = 0, \frac{k}{3^k} = 0$. Therefore:$\displaystyle \sum_{k=1}^n \frac{k}{3^k}=\sum_{k=0}^n \frac{k}{3^k}=V_n$ ... (1)
Then:$\displaystyle V_n+U_n = \sum_{k=0}^n \frac{k}{3^k}+\sum_{k=0}^n \frac{1}{3^k}$
$\displaystyle =\sum_{k=0}^n \frac{k+1}{3^k}$
$\displaystyle =3\sum_{k=0}^n \frac{k+1}{3^{k+1}}$
$\displaystyle =3\sum_{k=1}^{n+1} \frac{k}{3^{k}}$
$\displaystyle =3\sum_{k=0}^{n+1} \frac{k}{3^{k}}$, from (1)
$\displaystyle =3V_{n+1}$
Grandad