Video: AQA GCSE Mathematics Higher Tier Pack 2 β€’ Paper 1 β€’ Question 12

Two sequences have the expressions 3𝑛 + 2 and 4𝑛 βˆ’ 1 for the 𝑛th term. Find three numbers between 30 and 60 which are in both sequences.

05:12

Video Transcript

Two sequences have the expressions three 𝑛 plus two and four 𝑛 minus one for the 𝑛th term. Find three numbers between 30 and 60 which are in both sequences.

One method for solving this problem is to write out all the numbers in both expressions and find the common numbers between 30 and 60. The first sequence has an 𝑛th term expression three 𝑛 plus two.

This means that we can calculate the first term by substituting 𝑛 equals one. This gives us three multiplied by one plus two. Three multiplied by one is equal to three. And adding two gives us five. Therefore, the first term in this sequence is five.

The number in front of 𝑛, known as the coefficient, is the common difference. In this case, the common difference is three. This means that each term will be three more than the previous term.

Five plus three is equal to eight and eight plus three is equal to 11. This means that the second term in the sequence is eight and the third term is 11. We can continue the sequence by adding three each time all the way up to 60 as we require numbers between 30 and 60.

An alternative method for working out all the terms in the sequence would be substituting in 𝑛 equals two, 𝑛 equals three, and so on. For example, to work out the fifth term, 17, we could perform the calculation three multiplied by five plus two. Three multiplied by five is 15 and adding two is 17.

We now need to work out all the terms for the second sequence, four 𝑛 minus one. Once again, we can calculate the first term by substituting 𝑛 equals one. Four multiplied by one is equal to four and subtracting one gives us three. Therefore, the first term in this sequence is three.

The common difference this time is four as the coefficient of 𝑛 is equal to four. This means that each term will be four more than the previous term. Three plus four is equal to seven and seven plus four equals 11. Therefore, the second term in this sequence is seven and the third term is 11.

We can immediately see that 11 is a common term. It is in both sequences. However, this is not between 30 and 60. We must, therefore, continue the sequence by adding four up to 60. This gives us values 15, 19, 23, and so on up to 59.

Five of these numbers occur in both sequences: 11, 23, 35, 47, and 59. Of these, three are between 30 and 60: the numbers 35, 47, and 59. We can, therefore, say that the three numbers between 30 and 60 that are in the sequences three 𝑛 plus two and four 𝑛 minus one are 35, 47, and 59.

It is worth noting at this stage that the difference between each of the common terms is 12. 11 plus 12 equals 23, 23 plus 12 is 35, 35 plus 12 is 47, and 47 plus 12 is equal to 59. The common difference of sequence one was equal to three and the common difference of sequence two was equal to four.

The lowest common multiple of three and four is 12 as 12 is the lowest number in both the three and four times tables. This means that once we have found the first common number, in this case 11, we can keep adding 12 to find all the other common numbers.

The first sequence is going up in the three and the second sequence is going up in four. Therefore, both sequences will have the numbers going up in 12 from the first common number.

It is, therefore, clear that 11, 23, 35, 47, 59, and so on will be in both sequences, giving us the answers 35, 47, and 59 that are the numbers between 30 and 60 in both sequences.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.