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.

\( \begin{aligned} \displaystyle \require{color}
T_{1} &= 5 \\
d &= 6 \\
T_{n} &= 5+ (n-1)6 \\
&= 5+ 6n-6 \\
&= 6n-1 \\
6n-1 &\gt 500 \\
6n &\gt 501 \\
n &\gt 83.5 \\
T_{83} &= 6 \times 83 - 1 \\
&= 497 \\
T_{84} &= 6 \times 83 - 1 \\
&= 503 \\
\end{aligned} \\
\text{Therefore 503 is the first term of the sequence to exceed 500.} \\
\)

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?

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?

Insert three numbers between 2 and 22 so that all five numbers are in arithmetic sequence.

2, , , , 22

Correct

Incorrect

Question 2 of 5

2. Question

An arithmetic sequence starts \(24,28,32,36, \cdots \) . Find the first term of the sequence to exceed 2000.

Term

Correct

Incorrect

Question 3 of 5

3. Question

An arithmetic sequence starts \(9,16,23,30, \cdots \) . Find the first term of the sequence to exceed 500.

Term

Correct

Incorrect

Question 4 of 5

4. Question

A worker is employed at an initial salary of $22 000 per year. After each year of service he receives an increment of $1200 until he reaches the maximum salary of $35 200. How long does he have to work until he receives the maximum salary?

years

Correct

Incorrect

Question 5 of 5

5. Question

The sum of n terms of an arithmetic series is given by \({S_n} = 2{n^2}\) . Find the \({20^{{\rm{th}}}}\) term.