Sequences & Series

[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: 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]

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: 6]

The first term and the common ratio of a geometric series are denoted, respectively, by $u_1$ and $r$, where $u_1,r \in \mathbb{Q}$. Given that the fourth term is $64$ and the sum to infinity is $625$, find the value of $u_1$ and the value of $r$.

[Maximum mark: 6]

A bouncy ball is dropped from a height of $2$ metres onto a concrete floor. After hitting the floor, the ball rebounds back up to $80$ % of it's previous height, and this pattern continues on repeatedly, until coming to rest.

1. Show that the total distance travelled by the ball until coming to rest can be expressed by

   $2 + 4(0.8) + 4(0.8)^2 + 4(0.8)^3 + \cdots$[2]
   

2. Find an expression for the total distance travelled by the ball, in terms of the number of bounces, $n$. [2]
   

3. Find the total distance travelled by the ball. [2]
   

[Maximum mark: 6]

Given a sequence of integers, between $20$ and $300$, which are divisible by $9$.

1. Find their sum. [2]
   

2. Express this sum using sigma notation. [2]
   

An arithmetic sequence has first term $-500$ and common difference of $8$. The sum of the first $n$ terms of this sequence is negative.

3. Find the greatest value of $n$. [2]
   

[Maximum mark: 7]

The sides of a square are $8$ cm long. A new square is formed by joining the midpoints of the adjacent sides and two of the resulting triangles are shaded as shown. This process is repeated $5$ more times to form the right hand diagram below.

1. Find the total area of the shaded region in the right hand diagram above. [4]
   

2. Find the total area of the shaded region if the process is repeated indefinitely.[3]
   

[Maximum mark: 13]

1. The following diagram shows [PQ], with length $4$ cm. The line is divided into an infinite number of line segments. The diagram shows the first four segments.

   

   The length of the line segments are $m$ cm, $m^2$ cm, $m^3$ cm, $\dots$, where $0 < m < 1$.

   Show that $m = \dfrac{4}{5}$. [5]
   

2. The following diagram shows [RS], with length $l$ cm, where $l > 1$. Squares with side lengths $n$ cm, $n^2$ cm, $n^3$ cm, $\dots$, where $0 < n < 1$, are drawn along [RS]. This process is carried on indefinitely. The diagram shows the first four squares.

   

   The total sum of the areas of all the squares is $\dfrac{25}{11}$. Find the value of $l$. [8]