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?

Question

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.

Answer 1

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.

Answer 2

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?