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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.

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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.

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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?

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  • $\begingroup$ There are four increases from step 6 to step 10. $\endgroup$ – N. F. Taussig Mar 23 '15 at 12:49
  • $\begingroup$ 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. $\endgroup$ – elWanderero Mar 23 '15 at 12:57

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