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
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
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.