Q1
Answer
State if the following series converge. If they do, find the sum to infinity.
  1. $10-5+2.5...$
  2. $0.4+0.8+1.2...$
  1. $S_{\infty}=\frac{20}{3}$
  2. Diverges
Q2
Answer
By writing $0.\dot{2}\dot{3}$ as $\dfrac{23}{100}+\dfrac{23}{10000}+...$, find $0.\dot{2}\dot{3}$ as a fraction.
$\dfrac{23}{99}$
Q3
Answer
Find the sum of the first 50 even numbers.
$2550$
Q4
Answer
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 $b$.
$a=\dfrac{8}{3}b$
Q5
Answer
Given that the series $7+12+17+...$ exceeds $1000$, what is the fewer number of terms it has?
$20$
Q6
Answer
The sum to $20$ terms of an arithmetic series is $-15$, and the first term is $4$. Find the common difference.
$-\dfrac{1}{2}$
Q7
Answer
Find the exact sum of the following geometric series: $$\dfrac{2}{3}+\dfrac{4}{15}+...+\dfrac{32}{1875}$$
$\dfrac{2062}{1875}$
Q8
Answer
The sum of the geometric series $3+6+12...$ exceeds 1.5 million. What is the fewest number of terms it can have?
$19$
Q9
Answer
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.
$r = \pm \dfrac{2}{5}$
Q10
Answer
A teacher gives out biscuits in lessons. On the $n$th day, $5n+1$ biscuits are given out.
  1. How many biscuits has the teacher has given out by the end of day 10?
  2. The teacher started with $1029$ biscuits. How many days can they give the full amount of biscuits out on?
  1. $285$
  2. $19$
Q11
Answer
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 $n$.
$25$
Q12
Answer
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.
$\dfrac{5187}{8}$
Q13
Answer
For a geometric series, $S_3 = 9$ and $S_{\infty}=8$. Find the first term and common ratio.
$a = 12$, $r = -\dfrac{1}{2}$
Q14
Answer
For the geometric series $$1-2x+4x^2...$$
  1. Find the range of possible values for $x$ for which this is convergent.
  2. Find $S_{\infty}$ in terms of $x$.
  1. $-0.5 < x < 0.5$
  2. $\dfrac{1}{1+2x}$
Q15
Answer
A geometric series has first term $a$ and common ratio $r$. The second term is $\dfrac{15}{8}$, and $S_{\infty}=8$.
  1. One possible value of $r$ is $\frac{3}{8}$. Find the other possible value of $r$ and the corresponding value of $a$.
  2. Given that $r = \frac{3}{8}$, find the smallest possible value for $n$ such that $S_n > 7.99$.
  1. $r = \frac{5}{8}$, $a = 3$
  2. $7$
Q16
Answer
Write the following using Sigma notation:
  1. $2+4+6+8$
  2. $6+4.5+3+1.5+0-1.5$
  1. $\displaystyle\sum_{n=1}^4 2n$
  2. $\displaystyle\sum_{n=1}^6 -1.5n+7.5$
Q17
Answer
Write the following using Sigma notation: $$\dfrac{1}{3}+\dfrac{2}{15}+\dfrac{4}{75}+...+\dfrac{64}{46875}$$
$\displaystyle\sum_{n=1}^7 \dfrac{5}{6}\times\dfrac{2}{5}^n$
Q18
Answer
Evaluate:
  1. $\displaystyle\sum_{r=1}^{15} 3\times2^r$
  2. $\displaystyle\sum_{r=1}^{\infty} 7\times\left(-\dfrac{1}{3}\right)^r$
  1. $196602$
  2. $-\dfrac{7}{4}$
Q19
Answer
Evaluate: $$\displaystyle\sum_{r=9}^{30} 5r-\frac{1}{2}$$
$2134$
Q20
Answer
Evaluate: $$\displaystyle\sum_{r=5}^{100} 3\times0.5^r$$
$0.1875$
Q21
Answer
Find, in terms of $k$, $\displaystyle\sum_{r=10}^k 7-2r$
$6k-k^2+27$
Q22
Answer
Given that $\displaystyle\sum_{r=1}^k 8+3r = 377$, find $k$.
$13$
Q23
Answer
Given: $$\sum_{r=1}^nr^2 =\dfrac{1}{6}n(n+1)(2n+1)$$ evaluate: $$\displaystyle\sum_{r=3}^{12} 3r^2 + 4r$$
$2235$
Q24
Answer
Given that $$\displaystyle\sum_{r=1}^{10} a+(r-1)d = \sum_{r=11}^{14}a+(r-1)d$$ find $d$ in terms of $a$
$d=6a$