Topic 1 All

[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: 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]

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]

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: 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]

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]

1. Write down the value of

   1. $\log_2 8$;

   2. $\log_5\Big(\dfrac{1}{25}\Big)$;

   3. $\log_9 3$. [3]
      

2. Hence solve $\log_2 8 + \log_5\Big(\dfrac{1}{25}\Big) + \log_9 3 = \log_{16} x$.[3]
   

[Maximum mark: 7]

The first three terms of a geometric sequence are $u_1 = 0.8$, $u_2 = 2.4$, $u_3 = 7.2$.

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

2. Find the value of $S_8$. [2]
   

3. Find the least value of $n$ such that $S_n > 35\hspace{0.15em}000$. [3]
   

[Maximum mark: 6]

On $1$st of January $2022$, Grace invests $\$P$ in an account that pays a nominal annual interest rate of $6$ %, compounded quarterly.

The amount of money in Grace's account at the end of each year follows a geometric sequence with common ratio, $\alpha$.

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

Grace makes no further deposits or withdrawals from the account.

2. Find the year in which the amount of money in Grace's account will become triple the amount she invested. [3]
   

[Maximum mark: 6]

Jack rides his bike to work each morning. During the first minute, he travels $160$ metres. In each subsequent minute, he travels $80$ % of the distance travelled during the previous minute.

The distance from his home to work is $750$ metres. Jack leaves his house at $8$:$30$ am and must be at work at $8$:$40$ am.

Will Jack arrive to work on time? Justify your answer.

[Maximum mark: 5]

Find the values of $x$ when $25^{x^2-2x} = \left(\dfrac{1}{125}\right)^{4x+2}$.

[Maximum mark: 5]

The third term of an arithmetic sequence is equal to $7$ and the sum of the first $8$ terms is $20$.

Find the common difference and the first term.

[Maximum mark: 6]

The sum of the first three terms of a geometric sequence is $81.3$, and the sum of the infinite sequence is $300$. Find the common ratio.

[Maximum mark: 6]

Consider the expansion of $\bigg(x^3+\dfrac{2}{x}\bigg)^8$.

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

2. Find the coefficient of the term in $x^4$. [5]
   

[Maximum mark: 6]

The $1$st, $5$th and $13$th terms of an arithmetic sequence, with common difference $d$, $d \neq 0$, are the first three terms of a geometric sequence, with common ratio $r$, $r \neq 1$. Given that the $1$st term of both sequences is $12$, find the value of $d$ and the value of $r$.

[Maximum mark: 6]

Consider the expansion of $\bigg(3x + \dfrac{p}{x}\bigg)^8$, where $p > 0$. The coefficient of the term

in $x^4$ is equal to the coefficient of the term in $x^6$. Find $p$.

[Maximum mark: 8]

It is known that the number of trees in a small forest will decrease by $5$ % each year unless some new trees are planted. At the end of each year, $600$ new trees are planted to the forest. At the start of $2021$, there are $8200$ trees in the forest.

1. Show that there will be roughly $9060$ trees in the forest at the start of $2026$. [4]
   

2. Find the approximate number of trees in the forest at the start of $2041$. [4]
   

[Maximum mark: 14]

The first two terms of an infinite geometric sequence, in order, are

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

2. Show that the sum of the infinite sequence is $9\log_3 x$. [3]
   

The first three terms of an arithmetic sequence, in order, are

3. Find the common difference $d$, giving your answer as an integer. [3]
   

Let $S_6$ be the sum of the first $6$ terms of the arithmetic sequence.

4. Show that $S_6 = 6\log_3 x - 15$. [3]
   

5. Given that $S_6$ is equal to one third of the sum of the infinite geometric
   sequence, find $x$, giving your answer in the form $a^p$ where $a,p \in \mathbb{Z}$. [3]
   

[Maximum mark: 15]

The first three terms of an infinite geometric sequence are $k-4,\,\, 4,\,\, k+2$, where $k \in \mathbb{Z}$.

1. 1. Write down an expression for the common ratio, $r$.

   2. Hence show that $k$ satisfies the equation $k^2 - 2k - 24 = 0$.[5]
      

2. 1. Find the possible values for $k$.

   2. Find the possible values for $r$. [5]
      

3. The geometric sequence has an infinite sum.

   1. Which value of $r$ leads to this sum. Justify your answer.

   2. Find the sum of the sequence. [5]
      

[Maximum mark: 7]

Given that $(5+nx)^2\bigg(1+\dfrac{3}{5}x\bigg)^n\hspace{-0.25em}=\hspace{0.05em}25+100x+\cdots$, find the value of $n$.