Topic 1 All - Number & Algebra

[Maximum mark: 4]

Expand $(2x + 1)^4$ in descending powers of $x$ and simplify your answer.

[Maximum mark: 6]

An arithmetic sequence has $u_1= 40$, $u_2 = 32$, $u_3 = 24$.

1. Find the common difference, $d$. [2]
   

2. Find $u_8$. [2]
   

3. Find $S_8$. [2]
   

[Maximum mark: 7]

Find the value of each of the following, giving your answer as an integer.

1. $\log_{10} 100$. [2]
   

2. $\log_{10} 50 + \log_{10} 2$. [2]
   

3. $\log_{10} 4 - \log_{10} 40$. [3]
   

[Maximum mark: 6]

Find the value of each of the following, giving your answer as an integer.

1. $\log_6 6$. [2]
   

2. $\log_6 9 + \log_6 4$. [2]
   

3. $\log_6 72 - \log_6 2$. [2]
   

[Maximum mark: 6]

Consider an arithmetic sequence $2,6,10,14,\dots$

1. Find the common difference, $d$. [2]
   

2. Find the $10$th term in the sequence. [2]
   

3. Find the sum of the first $10$ terms in the sequence. [2]
   

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

1. State which sequence is arithmetic and find the common difference of the sequence. [2]
   

2. State which sequence is geometric and find the common ratio of the sequence.[2]
   

3. For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]
   

[Maximum mark: 6]

Consider the infinite geometric sequence $4480$, $-3360$, $2520$, $-1890,\dots$

1. Find the common ratio, $r$. [2]
   

2. Find the $20$th term. [2]
   

3. Find the exact sum of the infinite sequence. [2]
   

[Maximum mark: 4]

Expand $(2x - 3)^4$ in descending powers of $x$ and simplify your answer.

[Maximum mark: 4]

Prove that the sum of three consecutive positive integers is divisible by $3$.

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

1. State which sequence is arithmetic and find the common difference of the sequence. [2]
   

2. State which sequence is geometric and find the common ratio of the sequence.[2]
   

3. For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction. [2]
   

[Maximum mark: 4]

Consider two consecutive positive integers, $k$ and $k+1$.

Show that the difference of their squares is equal to the sum of the two integers.

[Maximum mark: 6]

The table shows the first four terms of three sequences: $u_n$, $v_n$, and $w_n$.

1. State which sequence is

   1. arithmetic;

   2. geometric. [2]
      

2. Find the sum of the first $50$ terms of the arithmetic sequence. [2]
   

3. Find the exact value of the $13$th term of the geometric sequence. [2]
   

[Maximum mark: 4]

The product of three consecutive integers is increased by the middle integer.

Prove that the result is a perfect cube.

[Maximum mark: 6]

Consider the expansion of $(2x-1)^9$.

1. Write down the number of terms in this expansion. [1]
   

2. Find the coefficient of the term in $x^2$. [5]
   

[Maximum mark: 6]

Jeremy invests $\$8000$ into a savings account that pays an annual interest rate of $5.5$ %, compounded annually.

1. Write down a formula which calculates that total value of the investment after $n$ years. [2]
   

2. Calculate the amount of money in the savings account after:

   1. $1$ year;

   2. $3$ years. [2]
      

3. Jeremy wants to use the money to put down a $\$10\hspace{0.15em}000$ deposit on an apartment. Determine if Jeremy will be able to do this within a $5$-year timeframe.[2]
   

[Maximum mark: 5]

Consider the expansion of $(x-3)^8$.

1. Write down the number of terms in this expansion. [1]
   

2. Find the coefficient of the term in $x^6$. [4]
   

[Maximum mark: 6]

Consider the infinite geometric sequence $9000$, $-7200$, $5760$, $-4608$, ...

1. Find the common ratio. [2]
   

2. Find the $25$th term. [2]
   

3. Find the exact sum of the infinite sequence. [2]
   

[Maximum mark: 6]

Let $\log_2 a = p$, $\log_2 b = q$, $\log_2 c = r$. Write down the following expressions in terms of $p$, $q$ and $r$.

1. $\log_2\Big(\dfrac{ab}{c}\Big)$ [2]
   

2. $\log_2\Big(\dfrac{a^2c}{b^3}\Big)$ [2]
   

3. $\log_a b$ [2]
   

[Maximum mark: 7]

An arithmetic sequence is given by $3$, $5$, $7,\dots$

1. Write down the value of the common difference, $d$. [1]
   

2. Find

   1. $u_{10}$;

   2. $S_{10}$. [4]
      

3. Given that $u_n = 253$, find the value of $n$. [2]
   

[Maximum mark: 7]

Let $p=\ln 2$ and $q = \ln 6$. Write down the following expressions in terms of $p$ and $q$.

1. $\ln 12$ [2]
   

2. $\ln 3$ [2]
   

3. $\ln 48$ [3]
   

[Maximum mark: 6]

Consider the following sequence of figures.

Figure 1 contains $6$ line segments.

1. Given that Figure $n$ contains $101$ line segments, show that $n = 20$.[3]
   

2. Find the total number of line segments in the first $20$ figures. [3]
   

[Maximum mark: 6]

Hannah buys a car for $\$24\hspace{0.15em}900$. The value of the car depreciates by $16$ % each year.

1. Find the value of the car after $10$ years. [3]
   

Patrick buys a car for $\$12\hspace{0.15em}000$. The car depreciates by a fixed percentage each year, and after $6$ years it is worth $\$6200$.

2. Find the annual rate of depreciation of the car. [3]
   

[Maximum mark: 6]

A $3$D printer builds a set of $49$ Ei$\text{f}$fel Tower Replicas in different sizes. The height of the largest tower in this set is $64$ cm. The heights of successive smaller towers are $95$ % of the preceding larger tower, as shown in the diagram below.

1. Find the height of the smallest tower in this set. [3]
   

2. Find the total height if all $49$ towers were placed one on top of another. [3]
   

[Maximum mark: 6]

Let $a = \log_5b$, where $b > 0$. Write down each of the following expressions
in terms of $a$.

1. $\log_5b^4$ [2]
   

2. $\log_5 (25b)$ [2]
   

3. $\log_{25}b$ [2]
   

[Maximum mark: 6]

Julia wants to buy a house that requires a deposit of $74\hspace{0.15em}000$ Australian dollars (AUD).

Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of $5.5$ %, compounded monthly.

1. Find the amount of AUD Julia needs to invest to reach $74\hspace{0.15em}000$ AUD after $8$ years. Give your answer correct to the nearest dollar. [3]
   

Julia's parents offer to add $5000$ AUD to her initial investment from part (a), however, only if she invests her money in a more reliable bank that pays a nominal annual interest rate only of $3.5$ %, compounded quarterly.

2. Find the number of years it would take Julia to save the $74\hspace{0.15em}000$ AUD if she accepts her parents money and follows their advice. Give your answer correct to the nearest year. [3]
   

[Maximum mark: 6]

The fifth term, $u_5$, of a geometric sequence is $125$. The sixth term, $u_6$, is $156.25$.

1. Find the common ratio of the sequence. [2]
   

2. Find $u_1$, the first term of the sequence. [2]
   

3. Calculate the sum of the first $12$ terms of the sequence. [2]
   

[Maximum mark: 6]

In an arithmetic sequence, $u_5 = 24$, $u_{13} = 80$.

1. Find the common difference. [2]
   

2. Find the first term. [2]
   

3. Find the sum of the first $20$ terms in the sequence. [2]
   

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Mia deposits $4000$ Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of $6$ %, compounded semi-annually.

1. Find the amount of interest that Mia will earn over the next $2.5$ years. [3]
   

Ella also deposits AUD into a bank account. Her bank pays a nominal annual $\text{interest}$ rate of $4$ %, compounded monthly. In $2.5$ years, the total amount in Ella's account will be $4000$ AUD.

2. Find the amount that Ella deposits in the bank account. [3]
   

[Maximum mark: 5]

Maria invests $\$25\hspace{0.15em}000$ into a savings account that pays a nominal annual interest rate of $4.25$%, compounded monthly.

1. Calculate the amount of money in the savings account after $3$ years. [2]
   

2. Calculate the number of years it takes for the account to reach $\$40\hspace{0.15em}000$. [3]
   

[Maximum mark: 6]

The first three terms of a geometric sequence are $u_1 = 32$, $u_2 = -16$, $u_3 = 8$.

1. Find the value of the common ratio, $r$. [2]
   

2. Find $u_6$. [2]
   

3. Find $S_{\infty}$. [2]
   

[Maximum mark: 6]

In an arithmetic sequence, $u_4 = 12$, $u_{11} = -9$.

1. Find the common difference. [2]
   

2. Find the first term. [2]
   

3. Find the sum of the first $11$ terms in the sequence. [2]
   

[Maximum mark: 6]

Emily deposits $2000$ Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of $4$ %, compounded monthly.

1. Find the amount of money that Emily will have in her bank account after $5$ years. Give your answer correct to two decimal places. [3]
   

Emily will withdraw the money back from her bank account when the amount reaches $3000$ AUD.

2. Find the time, in months, until Emily withdraws the money from her bank account. [3]
   

[Maximum mark: 6]

1. Show that $(2n-1)^3 + (2n+1)^3 = 16n^3+12n$ for $n \in \mathbb{Z}$. [3]
   

2. Hence, or otherwise, prove that the sum of the cubes of any two consecutive odd integers is divisible by four. [3]
   

[Maximum mark: 6]

A tennis ball bounces on the ground $n$ times. The heights of the bounces, $h_1, h_2, h_3, \dots,h_n,$ form a geometric sequence. The height that the ball bounces the first time, $h_1$, is $80$ cm, and the second time, $h_2$, is $60$ cm.

1. Find the value of the common ratio for the sequence. [2]
   

2. Find the height that the ball bounces the tenth time, $h_{10}$. [2]
   

3. Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to $2$ decimal places. [2]
   

[Maximum mark: 5]

In an arithmetic sequence, the sum of the 2nd and 6th term is $32$.
Given that the sum of the first six terms is $120$, determine the first term and common difference of the sequence.

[Maximum mark: 6]

Ali bought a car for $\$18\hspace{0.15em}000$. The value of the car depreciates by $10.5$ % each year.

1. Find the value of the car at the end of the first year. [2]
   

2. Find the value of the car after $4$ years. [2]
   

3. Calculate the number of years it will take for the car to be worth exactly half its original value. [2]
   

[Maximum mark: 5]

1. Prove that $\dfrac{5}{x^2} = \dfrac{5}{x(x-2)}-\dfrac{10}{x^2(x-2)}$. [3]
   

2. Determine the set of numbers $x$ for which the proof in part (a) is valid. [2]
   

[Maximum mark: 6]

The fourth term, $u_4$, of a geometric sequence is $135$. The fifth term, $u_5$, is $81$.

1. Find the common ratio of the sequence. [2]
   

2. Find $u_1$, the first term of the sequence. [2]
   

3. Calculate the sum of the first $20$ terms of the sequence. [2]
   

[Maximum mark: 5]

An arithmetic sequence has first term $45$ and common difference $-1.5$.

1. Given that the $k$th term of the sequence is zero, find the value of $k$. [2]
   

Let $S_n$ denote the sum of the first $n$ terms of the sequence.

2. Find the maximum value of $S_n$. [3]
   

[Maximum mark: 6]

The fifth term, $u_5$, of an arithmetic sequence is $25$. The eleventh term, $u_{11}$, of the same sequence is $49$.

1. Find $d$, the common difference of the sequence. [2]
   

2. Find $u_1$, the first term of the sequence. [2]
   

3. Find $S_{100}$, the sum of the first $100$ terms of the sequence. [2]
   

[Maximum mark: 6]

The third term, $u_3$, of an arithmetic sequence is $7$. The common difference of
the sequence, $d$, is $3$.

1. Find $u_1$, the first term of the sequence. [2]
   

2. Find $u_{60}$, the $60$th term of sequence. [2]
   

The first and fourth terms of this arithmetic sequence are the first two terms
of a geometric sequence.

3. Calculate the sixth term of the geometric sequence. [2]
   

[Maximum mark: 5]

The third term, in descending powers of $x$, in the expansion of $(x+p)^8$ is $252x^6$. Find the possible values of $p$.

[Maximum mark: 6]

Greg has saved $2000$ British pounds (GBP) over the last six months. He decided to deposit his savings in a bank which offers a nominal annual interest rate of $\text{\(8\)\hspace{0.05em}\%}$, compounded monthly, for two years.

1. Calculate the total amount of interest Greg would earn over the two years. Give your answer correct to two decimal places. [3]
   

Greg would earn the same amount of interest, compounded semi-annually, for two years if he deposits his savings in a second bank.

2. Calculate the nominal annual interest rate the second bank offers. [3]
   

[Maximum mark: 6]

On $1$st of January $2021$, Fiona decides to take out a bank loan to purchase a new Tesla electric car. Fiona takes out a loan of $\$P$ with a bank that offers a nominal annual interest rate of $2.6\hspace{0.05em}\%$, compounded monthly.

The size of Fiona's loan at the end of each year follows a geometric sequence with common ratio, $\alpha$.

1. Find the value of $\alpha$, giving your answer to five significant figures. [3]
   

The bank lets the size of Fiona's loan increase until it becomes triple the size of the original loan. Once this happens, the bank demands that Fiona pays the entire amount back to close the loan.

2. Find the year during which Fiona will need to pay back the loan. [3]
   

[Maximum mark: 6]

The first term of an arithmetic sequence is $24$ and the common difference is $16$.

1. Find the value of the $62$nd term of the sequence. [2]
   

The first term of a geometric sequence is $8$. The $4$th term of the geometric sequence is equal to the $13$th term of the arithmetic sequence given above.

2. Write down an equation using this information. [2]
   

3. Calculate the common ratio of the geometric sequence. [2]
   

[Maximum mark: 6]

The Australian Koala Foundation estimates that there are about $45\hspace{0.15em}000$ koalas left in the wild in $2019$. A year before, in $2018$, the population of koalas was estimated as $50\hspace{0.15em}000$. Assuming the population of koalas continues to decrease by the same percentage each year, find:

1. the exact population of koalas in $2022$; [3]
   

2. the number of years it will take for the koala population to reduce to half of its number in $2018$. [3]