Sequences & Series

[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]

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]

Sarah walks to school each morning. During the first minute, she travels $130$ $\text{metres}$. In each subsequent minute, she travels $5$ metres less than the distance she travelled during the previous minute. The distance from her home to school is $950$ metres. Sarah leaves her house at $8$:$00$ am and must be at school by $8$:$10$ am.

Will Sarah arrive to school on time? Justify your answer.

[Maximum mark: 7]

The first three terms of a geometric sequence are $u_1 = 0.4$, $u_2 = 0.6$, $u_3 = 0.9$.

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

2. Find the sum of the first ten terms in the sequence. [2]
   

3. Find the greatest value of $n$ such that $S_n < 650$. [3]
   

[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: 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]
   

[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]