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Calculate the number of terms in the arithmetic sequence

$$a, a + d, a + 2d,\cdots, a + (n-1)d.$$

I don't have a problem answering this question with an integer sequence but I'm a bit lost with what to do for the general formula.

Allawonder
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Dan A
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    How many $d$'s does the first term have? The second term? The $n$th term? – Andrew Chin Sep 27 '19 at 21:25
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    A variation on @AndrewChin's suggestion. What happens when $n=1$? $n=2$? $n=3$? Very often when you are stuck looking for some kind of pattern, working with the first few cases gives a big clue. Sometimes you just need to be careful because the first cases occasionally have special features and you need to go a few in before things become clear. In many cases it is quite cheap - takes little time and effort, so it is worth a try. – Mark Bennet Sep 27 '19 at 21:42

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The $a$ is fixed, so it doesn't tell us anything about the number of terms. Thus, you may neglect it. What you watch is the coefficient of $d,$ which varies, beginning from $0,$ and increasing by $1$ each step of the way.

In other words you only have to deal with the simpler progression $$0,1,2,3,\cdots, n-2, n-1.$$ Can you tell how many terms there are now?

Allawonder
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