Investigations (Cambridge (CIE) IGCSE International Maths: Core): Exam Questions

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.

You may use your answer from question 1(b).

Is it true that all triangular numbers are hexagonal numbers?

Explain your answer.

Based on the results above, which triangular number out of or is more likely to be a hexagonal number?

Give a reason for your answer.

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

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:

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.

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

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.

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 shown below.

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

A new sequence, , is formed by summing each column in the table above.

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

Find , and .

Using any information from the table in question 4, find .

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

Find a formula for the th term of the sequence .

Find the value of .

The th term formula for a hexagonal number, , is given by

For example, to find the third hexagonal number:

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

If the th hexagonal number, , is equal to 780, use the th term formula to show that must satisfy the equation

Show that 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 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:

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.

and

Find the value of .

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

The prime number creates a perfect number.

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