1. State if the following series converge. If they do, find the sum to infinity.
$10-5+2.5...$
$0.4+0.8+1.2...$
$S_{\infty}=\frac{20}{3}$
Diverges
2. By writing $0.\dot{2}\dot{3}$ as $\dfrac{23}{100}+\dfrac{23}{10000}+...$, find $0.\dot{2}\dot{3}$ as a fraction.
$\dfrac{23}{99}$
3. Find the sum of the first 50 even numbers.
$2550$
4. Two geometric series have first terms $a$ and $b$ and common ratios $2$ and $3$ respectively. The sum of the first four terms of these series are equal. Find $a$ in terms of $b$.
$a=\dfrac{8}{3}b$
5. Given that the series $7+12+17+...$ exceeds $1000$, what is the fewer number of terms it has?
$20$
6. The sum to $20$ terms of an arithmetic series is $-15$, and the first term is $4$. Find the common difference.
$-\dfrac{1}{2}$
7. Find the exact sum of the following geometric series: $$\dfrac{2}{3}+\dfrac{4}{15}+...+\dfrac{32}{1875}$$
$\dfrac{2062}{1875}$
8. The sum of the geometric series $3+6+12...$ exceeds 1.5 million. What is the fewest number of terms it can have?
$19$
9. The sum of the first two terms of a geometric series is $4.48$. The sum of the first four terms is $5.1968$. Find two possible values for the common ratio of this series.
$r = \pm \dfrac{2}{5}$
10. A teacher gives out biscuits in lessons. On the $n$th day, $5n+1$ biscuits are given out.
How many biscuits has the teacher has given out by the end of day 10?
The teacher started with $1029$ biscuits. How many days can they give the full amount of biscuits out on?
$285$
$19$
11. The fifth term of an arithmetic series is $33$ and the tenth term is $68$. The sum of the first $n$ terms is $2225$. Find $n$.
$25$
12. The first three terms of a geometric series are $$(k-6) \quad k \quad (2k+5)$$ where $k$ is a positive constant. Find the exact sum of the first 6 terms of this series.
$\dfrac{5187}{8}$
13. For a geometric series, $S_3 = 9$ and $S_{\infty}=8$. Find the first term and common ratio.
$a = 12$, $r = -\dfrac{1}{2}$
14. For the geometric series $$1-2x+4x^2...$$
Find the range of possible values for $x$ for which this is convergent.
Find $S_{\infty}$ in terms of $x$.
$-0.5 < x < 0.5$
$\dfrac{1}{1+2x}$
15. A geometric series has first term $a$ and common ratio $r$. The second term is $\dfrac{15}{8}$, and $S_{\infty}=8$.
One possible value of $r$ is $\frac{3}{8}$. Find the other possible value of $r$ and the corresponding value of $a$.
Given that $r = \frac{3}{8}$, find the smallest possible value for $n$ such that $S_n > 7.99$.
$r = \frac{5}{8}$, $a = 3$
$7$
16. Write the following using Sigma notation:
$2+4+6+8$
$6+4.5+3+1.5+0-1.5$
$\displaystyle\sum_{n=1}^4 2n$
$\displaystyle\sum_{n=1}^6 -1.5n+7.5$
17. Write the following using Sigma notation: $$\dfrac{1}{3}+\dfrac{2}{15}+\dfrac{4}{75}+...+\dfrac{64}{46875}$$
