# A sequence $u_1,u_2,u_3,\dots$ is defined by $u_n=10 \times 0.6^n$

A sequence $u_1,u_2,u_3,\dots$ is defined by $u_n=10 \times 0.6^n$. Find $$\sum_{n=1}^\infty u_n$$

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Take the geometric series $$\sum_{i=0}^n u^i= \frac{u^{n+1}-1}{u-1}$$ and try to write your sum in one of those
As you have $$\sum_{n=0}^\infty 10 \cdot 0.6^n -10 = 10 \cdot \sum_{n=0}^\infty 0.6^n-10= 10 \cdot \frac{1}{1-0.6}-10=10\cdot \frac{1}{0.4}-10=25-10=15$$
• @moamen: You’re welcome. (You can start learning how to do that kind of editing here.) Yes, it’s $\frac{a}{1-r}$, where $a$ is the first term, and $r$ is the ratio of consecutive terms. – Brian M. Scott Mar 25 '13 at 20:39
• @moamen: Not quite: your summation starts at $n=1$, so $a=u_1=10\cdot0.6=6$. – Brian M. Scott Mar 25 '13 at 20:49