An arithmetic sequence has 11 terms. Which is the middle term?
We recall that the general term of an arithmetic sequence, written 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. In this sequence, we are told we have 11 terms from 𝑎 sub one to 𝑎 sub 11. One way of finding the middle term would be to cross off from either end. We cross off 𝑎 sub one and 𝑎 sub 11. Our next pair is 𝑎 sub two and 𝑎 sub 10. Continuing this pattern, we can cross off another three terms from either end, leaving us with the middle term 𝑎 sub six.
Alternatively, we could use the fact that if 𝑎 sub 𝑚 is the middle term, then 𝑚 is equal to 𝑛 plus one divided by two, where 𝑛 is the total number of terms, in this case 11. We need to add 11 and one and then divide the answer by two. 12 divided by two is equal to six, which confirms that the middle term will be 𝑎 sub six.