Question 1094753
<br>A key concept in working with arithmetic sequences and series is that you can always group the terms in pairs so that the sum in each pair is the same; or, if the number of terms in the sequence is odd, there will be a single term in the middle that is half of that common sum.<br>
So here is how I would work this problem....<br>
The sum of the first 6 terms is 78.  That means there are 3 pairs of terms, with each pair having a sum of 78/3 = 26.<br>
The 6th term is 23; it pairs up with the first term; and the sum of the first and 6th terms is 26.  That means the first term is 26-23 = 3.<br>
The 6th term, 23, is the first term, 3, plus the common difference 5 times:
{{{23 = 3 + 5d}}}
{{{20 = 5d}}}
{{{d = 4}}}<br>
We are done with part (a): the common difference is 4; the first term is 3.<br>
Part (b) is now easy.  The 10th term is the first term, plus the common difference 9 times:
{{{3 + 9(4) = 3+36 = 39}}}