Hi!
Could you help me with this assignment?
What is the sum of $\displaystyle n$ first elements of the sequence $\displaystyle (a_n)$where:
$\displaystyle a_1=2, a_2=22, a_3=222, ...$ ?
Thanks.
First you should have tried to find a formula for $\displaystyle a_{n}$. It's easy to observe that:
$\displaystyle a_{n}=2(10^{n-1}+10^{n-2}+...+10^{1}+10^{0})=2\cdot \frac{10^{n}-1}{10-1}=\frac{2}{9}\cdot (10^{n}-1)$
Now that you "know" $\displaystyle a_{n}$ it's simpler to calculate the sum:
$\displaystyle a_{1}+a_{2}+...+a_{n-1}+a_{n}=\frac{2}{9}\left ( 10^{1}+10^{2}+...+10^{n-1}+10^{n} -n\right )=\frac{2}{9}\left ( 10\cdot \frac{10^{n}-1}{10-1} -n \right )$
$\displaystyle S_{n}=\frac{2}{9}\cdot \frac{10^{n+1}-10-9n}{9}=\frac{2}{81}\cdot \left (10^{n+1}-9n-10 \right )$
Pretty clear I hope...