Investigations (Cambridge International IGCSE Maths): Exam Questions

This investigation introduces and explores the properties and patterns of the sequence of hexagonal numbers.

Hexagonal numbers are numbers formed by summing equally spaced dots around hexagons that sit inside of each other, as shown below.

If $n$ is the position of the term in the sequence then $H_{n}$ is the $n$th hexagonal number.

The first hexagonal number is 1 and the second hexagonal number is 6.

Use information on the diagram to write down the third hexagonal number.

Extract:

Hexagonal numbers are numbers formed by summing equally spaced dots around hexagons that sit inside of each other, as shown below. If n is the position of the term in the sequence then H_n is the nth hexagonal number. The diagram shows hexagonal numbers from n=1 to n=6, showing black dots in hexagon patterns. The first hexagonal number is 1 and the second hexagonal number is 6.

By counting the total number of dots in the diagram for $n=4$, work out the value of the fourth hexagonal number, $H_{4}$.

Extract:

If n is the position of the term in the sequence then H_n is the nth hexagonal number.

The next hexagonal number after 66, not shown on the diagram, is 91.

Write down the value of $n$ for this term.

The hexagonal numbers are shown above.

The first six triangular numbers are shown below.

Show that the first three hexagonal numbers are all triangular numbers.

Explain whether or not the fourth hexagonal number is also a triangular number.

Extract:

The hexagonal numbers are shown above. The first six triangular numbers are shown below.

Is it true that all triangular numbers are hexagonal numbers?

Explain your answer.

Extract:

Is it true that all triangular numbers are hexagonal numbers?

Explain your answer.

Based on the results above, which triangular number out of $T_{150}$ or $T_{151}$ is more likely to be a hexagonal number?

Give a reason for your answer.

The hexagonal numbers are a sequence of numbers. The first six hexagonal numbers are

$H_{1}=1,H_{2}=6,H_{3}=15,H_{4}=28,H_{5}=45,H_{6}=66$

To test whether a number is a hexagonal number or not, substitute it into the formula:

$\frac{1 + \sqrt{8 \times ( \text{number} ) + 1} }{4}$

If the output is a whole number, then it is a hexagonal number.

For example, to test if 6 is a hexagonal number, substitute it into the formula:

$\begin{gathered} \frac{1 + \sqrt{8 \times 6 + 1} }{4} & = \frac{1 + \sqrt{48 + 1} }{4} \\ = \frac{1 + \sqrt{49} }{4} \\ = \frac{1 + 7}{4} \\ = \frac{8}{4} \\ = 2 \end{gathered}$

The output is 2, which is a whole number, so 6 is a hexagonal number.

Use the test above to determine whether 120 is a hexagonal number.

You must show your working clearly.

Extract:

To test whether a number is a hexagonal number or not, substitute it into the formula:

$[\frac{1 + \sqrt{8 \times ( \text{number} ) + 1}}{4}]$

If the output is a whole number, then it is a hexagonal number.

For example, to test if 6 is a hexagonal number, substitute it into the formula:

$[ \begin{gathered} \frac{1 + \sqrt{8 \\times 6 + 1} }{4} & = \frac{1 + \sqrt{48 + 1} }{4} \\ = \frac{1 + \sqrt{49} }{4} \\ = \frac{1 + 7}{4} \\ = \frac{8}{4} \\ = 2 \end{gathered} ]$

The output is 2, which is a whole number, so 6 is a hexagonal number.

Use the test to show that 236 cannot be a hexagonal number.

The hexagonal numbers are a sequence of numbers. The 23^rd to the 26^th hexagonal numbers are shown in the table below.

A student believes that hexagonal numbers cannot be divisible by the prime number 17.

Use any information from the table to show that the student is not correct.

Extract:

The 23rd to the 26th hexagonal numbers are shown in the table below.

| n | ... | 23 | 24 | 25 | 26 | ... | | H_n | ... | 1035 | 1128 | 1225 | 1326 | ... |

Another student believes that the number 1 is the only square number that is also a hexagonal number.

Use any information from the table to show that this student is not correct.

The hexagonal numbers are a sequence of numbers. The first six hexagonal numbers are shown below.

A new sequence, $K_{n}$, is formed by summing each column in the table above.

The first three terms, $K_{1}$, $K_{2}$ and $K_{3}$, are shown in the table below.

Find $K_{4}$, $K_{5}$ and $K_{6}$.

The 23^rd hexagonal number, $H_{23}$, is 1035.

Find $K_{23}$.

Extract:

A new sequence, ![K subscript n], is formed by summing each column in the table above.

The first three terms, ![K subscript 1], ![K subscript 2] and ![K subscript 3], are shown in the table below.

| ![n] | 1 | 2 | 3 | 4 | 5 | 6 | | ![H subscript n] | 1 | 6 | 15 | | 45 | 66 | | ![K subscript n] | 2 | 8 | 18 | | | |

Describe in words how the new sequence, $K_{n}$, is related to the sequence of square numbers: 1, 4, 9, 16, 25, ...

Extract:

The hexagonal numbers are shown below.

The missing value, H_4, is your answer to question 1(b).

| n | 1 | 2 | 3 | 4 | 5 | 6 | | H_n | 1 | 6 | 15 | | 45 | 66 | A new sequence, K_n, is formed by summing each column in the table above.

The first three terms, K_1, K_2 and K_3, are shown in the table below.

| n | 1 | 2 | 3 | 4 | 5 | 6 | | H_n | 1 | 6 | 15 | | 45 | 66 | | K_n | 2 | 8 | 18 | | | | Find K_4, K_5 and K_6.

Find a formula for the $n$th term of the sequence $K_{n}$.

Extract:

Find a formula for the ![n]th term of the sequence ![K subscript n].

Find the value of $K_{11}$.

The hexagonal numbers are a sequence of numbers. The $n$th term formula for a hexagonal number, $H_{n}$, is given by

$H_{n} = n ( 2 n - 1 )$

For example, to find the third hexagonal number:

$\begin{gathered} H_{3} & = 3 ( 2 \times 3 - 1 ) \\ = 3 ( 6 - 1 ) \\ = 3 \times 5 \\ = 15 \end{gathered}$

Use the formula to find the 14^th hexagonal number.

Extract:

The nth term formula for a hexagonal number, H_n, is given by

H_n = n(2n - 1)

If the $n$th hexagonal number, $H_{n}$, is equal to 780, use the $n$th term formula to show that $n$ must satisfy the equation

$2n^{2}-n-780=0$

Extract:

2 n squared minus n minus 780 equals 0

Show that $n=20$ satisfies the equation in part (b).

A number is called a perfect number if it is equal to the sum of all its possible factors (including the factor 1, but not including the factor of the number itself).

For example, 6 is a perfect number, as its possible factors (not including 6) are 1, 2 and 3, which sum to give 6:

$1+2+3=6$

The hexagonal numbers are a sequence of numbers. The first six hexagonal numbers are

$H_{1}=1,H_{2}=6,H_{3}=15,H_{4}=28,H_{5}=45,H_{6}=66$

A perfect hexagonal number is a number that is both perfect and a hexagonal number.

Is 28 a perfect hexagonal number?

Show your working clearly.

The number 496 is known to be a perfect hexagonal number.

There are nine possible factors of 496 (not including itself), which are given below.

$1,2,4,8,16,31,k,124$ and $248$

Find the value of $k$.

It is possible to create a perfect number by substituting certain prime numbers, $p$, into the formula:

$2^{p - 1} ( 2^{p} - 1 )$

The prime number $p=7$ creates a perfect number.

Show that this perfect number is the 64^th hexagonal number.

You may use the fact that the $n$th term formula for a hexagonal number, $H_{n}$, is given by $H_{n} = n ( 2 n - 1 )$.