Topic

# Applications

Pattern occurs naturally in many real-life situations; for example the addition of interest to finance institutions, plant spacing in a building and the stacking of logs in a pile. Two of the most common patterns are termed arithmetic and geometric sequences. Recognition of these patterns is important in analysing situations that occur normally in the real world.

## Practice Questions

### Question 1

Insert two numbers between $22$ and $37$ so that all four numbers are in arithmetic sequence.

### Question 2

The $2$nd term of an arithmetic sequence is $13$ and the $5$th term is $31$. Find the $17$th term of this sequence.

### Question 3

Insert three numbers between $3$ and $19$ so that all five numbers are in arithmetic sequence.

### Question 4

An arithmetic sequence starts $13, 16, 19, 22,\cdots$. Find the first term of the sequence to exceed $1000.$

### Question 5

An arithmetic sequence starts $5, 11, 17, 23,\cdots$. Find the first term of the sequence to exceed $500.$

### Question 6

An accountant is employed at an initial salary of $27 750$ per annum. After each year of service he receives an increment of $1050$ until he reaches the maximum salary of $37 200.$

(a)  What is his salary after eight years of service?

(b)  How long does he have to work until he receives the maximum salary?

### Question 7

In the construction of a $5$ km expressway a truck delivers materials from a base. After depositing each load, the truck returns to the base to collect the next load. The first load is deposited 200 m from the base, the second 350 m from the base, and the third 500 m from the base. Each subsequent load is deposited 150 m from the previous one.

(a)  How far is the $15^\text{th}$ load deposited from the base?

(b)  How many loads are deposited along the total length of the 5 km expressway?

### Question 8

The sum of n terms of an arithmetic series is given by ${S_n}=2n+3{n^2}$.

(a)  Find the $2^\text{nd}$ term.

(b)  Find the $20^\text{th}$ term.