What is the sum of the first 4 terms of the arithmetic sequence in which the 6th term is 8 and the 10th term is 13?

Can somebody help me figure out how to approach this problem and why the answer is 14.5? I already have the answer I'm just confused about how to approach these questions in general for future purposes. Thank you.

Use the formula for the $n$th term of an arithmetic sequence:

$$a_n=a+(n-1)d$$

where $a_n$ is the $n$th term, $a$ is the first term, and $d$ is the common difference.

You have two pieces of information, with two variables $a$ and $d$. (You are given $a_n$ and $n$ in each piece of information.) These simultaneous equations are easily solved.

Then use the sum formula

$$S_n=\frac n2[2a+(n-1)d]$$

to get the desired sum.

There are other ways to solve this problem, but this way is general and can be used for many sequence problems.

An arithmetic sequence increases by the same constant at each step. From term 6 to term 10, an interval of five terms, it has increased by $13-8=5$. What is the increment at each step? And what must the starting value have been for the sequence, with that increment at each step, to reach 8 by step 6?

• There are four increases from step 6 to step 10. – N. F. Taussig Mar 23 '15 at 12:49
• By "steps" I meant of course the five terms from 8 to 13. Very bad choice of wording in hindsight, since a "step" by any standard does indeed mean an increment between points. – elWanderero Mar 23 '15 at 12:57