Maths Week Scotland 2023 - Problem 6 - Sequences

CHRIS: This challenge is all about sequences. A sequence is an ordered list of numbers that follow a pattern. There are two parts to this question so listen carefully.

Part A.

A simple sequence can be made starting with three and adding four each time: three, seven, eleven, fifteen, nineteen and so on. What do the first six numbers of this sequence add up to?

And it goes up in fours. So we'd have a three, then a seven.

Part B.

This part is about a different sequence of the same type, but we don't have all the information. The first number in this sequence is five and the sum of the first six numbers is one hundred and twenty. Using the information given, can you work out what the second number in the sequence is?

Explain your answer.

And I'm not gonna tell you what it goes up in and so what I want to know, what would the second term in this sequence be?

PUPIL: Remember that you always add the same amount to get to the next number in the sequence.

PUPIL: Don't forget that each number is made by adding an amount to five.

PUPIL: Try using algebra to make an equation.

PUPIL: Can I use algebra to work this question out?

CHRIS: Yeah. Algebra would be a great way to attack this question.

PUPIL: Good luck.

How did you get on with the sequences problem?

Let's look at the answers together.

Part A.

For Part A, we had a simple sequence starting with three and adding four each time: three, seven, eleven, fifteen, nineteen…

We had to figure it out what the first six numbers of this sequence add up to.

The sixth number in this sequence is twenty three because we're adding four each time.

So the sum is three plus seven plus eleven plus fifteen plus nineteen plus twenty three, which equals seventy eight.

Anybody got any good ideas about what we could do?

Part B.

Now, we've been given a different sequence of the same type. We know the first number in the sequence is five and the sum of the first six numbers is one hundred and twenty and we had to work out the second number in the sequence.

We could use algebra to find the solution. If the first number is five and we call the amount we add on each time 'd' then the second number is five plus d, the third number is five plus two d, etc.

The sum of the six numbers is five plus five plus d plus five plus two d plus five plus three d plus five plus four d plus five plus five d, which is thirty plus fifteen d, but we know that this sum is one hundred twenty, so thirty plus fifteen d equals one hundred and twenty.We could take thirty from each side, leaving fifteen d equals ninety.

Now, we can divide each side by fifteen to work out that d equals six. We know the second number in the sequence is five plus d and now we know what the value of d is so we can work out that that second number is five plus six, which is eleven.

Great work.