ONE EIGHTH CONTEST 2025
Presbyterian Boys' Sec. School - 55 points
Oti Boateng SHS - 35 points
T.I. AMASS, Kumasi - 34 points
ONE EIGHTH CONTEST 2025
Our Lady of Grace SHS - 56 points
Bishop Herman College - 39 points
Ejisuman SHS - 27 points
ONE EIGHTH CONTEST 2025
GSTS - 50 points
Nkwatia Presbyterian SHS - 45 points
Ashaiman SHS - 28 points
ROUND 1
PREAMBLE
SERIES AND SEQUENCE
Find the infinite sum of the given series. Simplify your answer as a rational number.
FIRST QUESTION
$1+0.1+0.01+0.001+\ldots$
ANSWER: $\dfrac{10}{9}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
where $a$ is the first term and $r$ is the common ratio.
$a=1$
$r=\dfrac{U_2}{U_1}$
$r=\dfrac{0.1}{1}$
$r=0.1$
$S_{\infty}=\dfrac{1}{1-0.1}$
$S_{\infty}=\dfrac{1}{0.9}$
$S_{\infty}=\dfrac{10}{9}$
SECOND QUESTION
$20-2+0.2-0.02+0.002-\ldots$
ANSWER: $\dfrac{200}{11}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
where $a$ is the first term and $r$ is the common ratio.
$a=20$
$r=\dfrac{U_2}{U_1}$
$r=\dfrac{-2}{20}$
$r=-\dfrac{1}{10}$
$S_{\infty}=\dfrac{20}{1-\left(-\dfrac{1}{10}\right)}$
$S_{\infty}=\dfrac{20}{\dfrac{11}{10}}$
$S_{\infty}=\dfrac{200}{11}$
THIRD QUESTION
$10+2+0.4+0.08+\ldots$
ANSWER: $\dfrac{25}{2}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
where $a$ is the first term and $r$ is the common ratio.
$a=10$
$r=\dfrac{U_2}{U_1}$
$r=\dfrac{2}{10}$
$r=\dfrac{1}{5}$
$S_{\infty}=\dfrac{10}{1-\dfrac{1}{5}}$
$S_{\infty}=\dfrac{10}{\dfrac{4}{5}}$
$S_{\infty}=\dfrac{25}{2}$
PRACTICE QUESTIONS
Simplify each answer as a rational number.
1. Find the sum to infinity of $3+0.3+0.03+0.003+\ldots$. Give the answer as a rational number.
ANSWER: $\dfrac{10}{3}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
The first term is $a=3$.
$r=\dfrac{U_2}{U_1}$
$r=0.1$
$S_{\infty}=\dfrac{3}{1-\dfrac{1}{10}}$
$S_{\infty}=\dfrac{3}{\dfrac{9}{10}}$
$S_{\infty}=\dfrac{10}{3}$
2. Find the sum to infinity of $9-0.9+0.09-0.009+\ldots$. Give the answer as a rational number.
ANSWER: $\dfrac{90}{11}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
The first term is $a=9$.
$r=\dfrac{U_2}{U_1}$
$r=-0.1$
$S_{\infty}=\dfrac{9}{1-(-\dfrac{1}{10})}$
$S_{\infty}=\dfrac{9}{\dfrac{11}{10}}$
$S_{\infty}=\dfrac{90}{11}$
3. Find the sum to infinity of $2+0.2+0.02+0.002+\ldots$. Give the answer as a rational number.
ANSWER: $\dfrac{20}{9}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
The first term is $a=2$.
$r=\dfrac{U_2}{U_1}$
$r=0.1$
$S_{\infty}=\dfrac{2}{1-\dfrac{1}{10}}$
$S_{\infty}=\dfrac{2}{\dfrac{9}{10}}$
$S_{\infty}=\dfrac{20}{9}$
4. Find the sum to infinity of $7+0.7+0.07+0.007+\ldots$. Give the answer as a rational number.
ANSWER: $\dfrac{70}{9}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
The first term is $a=7$.
$r=\dfrac{U_2}{U_1}$
$r=0.1$
$S_{\infty}=\dfrac{7}{1-\dfrac{1}{10}}$
$S_{\infty}=\dfrac{7}{\dfrac{9}{10}}$
$S_{\infty}=\dfrac{70}{9}$
5. Find the sum to infinity of $4-0.4+0.04-0.004+\ldots$. Give the answer as a rational number.
ANSWER: $\dfrac{40}{11}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
The first term is $a=4$.
$r=\dfrac{U_2}{U_1}$
$r=-0.1$
$S_{\infty}=\dfrac{4}{1-(-\dfrac{1}{10})}$
$S_{\infty}=\dfrac{4}{\dfrac{11}{10}}$
$S_{\infty}=\dfrac{40}{11}$
6. Find the sum to infinity of $6+0.6+0.06+0.006+\ldots$. Give the answer as a rational number.
ANSWER: $\dfrac{20}{3}$
SOLUTION
$S_{\infty}=\dfrac{a}{1-r}$
The first term is $a=6$.
$r=\dfrac{U_2}{U_1}$
$r=0.1$
$S_{\infty}=\dfrac{6}{1-\dfrac{1}{10}}$
$S_{\infty}=\dfrac{6}{\dfrac{9}{10}}$
$S_{\infty}=\dfrac{20}{3}$