Find the order of the term whose
value is 112 in the sequence 17, 22, 27, 32, and so on.
We begin by recalling that the
order of a term in a sequence is its position. In this question, we are given the
sequence 17, 22, 27, 32, and so on. This means that the first term, 𝑎
sub one, is equal to 17. The second term, 𝑎 sub two, is
22. The third term, 𝑎 sub three, is
27, and so on. We need to find the order or
position of the term whose value is 112.
We know that our sequence is
arithmetic, as there is a common difference between consecutive terms. In any arithmetic sequence, the
general term 𝑎 sub 𝑛 is equal to 𝑎 sub one plus 𝑛 minus one multiplied by 𝑑,
where 𝑎 sub one is the first term and 𝑑 is the common difference. We can see from our sequence that
𝑎 sub one is equal to 17 and the common difference 𝑑 is five. Substituting these values into the
general equation gives us 17 plus 𝑛 minus one multiplied by five.
We want to find the value of 𝑛
such that this expression is equal to 112. Distributing the parentheses, we
have 17 plus five 𝑛 minus five is equal to 112. The left-hand side simplifies to
five 𝑛 plus 12. We can then subtract 12 from both
sides such that five 𝑛 is equal to 100. Finally, dividing through by five
gives us 𝑛 is equal to 20. And we can therefore conclude that
the order of the term whose value is 112 is 𝑎 sub 20. It is the 20th term in the sequence
17, 22, 27, 32, and so on.
It is worth noting that all the
even-number terms in this sequence end in two. This suggests that our answer of 𝑎
sub 20 is correct.