   SEARCH HOME Math Central Quandaries & Queries  Question from Don: How many terms in arithmetic sequence are there if the first term and the last term are 3&59 respectively in common difference is 4? Hi Don,

Let me try a smaller example where I know the correct answer. You have an arithmetic sequence that starts with 3 and increases in steps of 4. Hence the sequence starts

3, 7, 11, 15, 19.

Suppose the question is

How many terms in arithmetic sequence are there if the first term and the last term are 3 & 19 respectively in common difference is 4?

You know the answer is 5 but how do you know that without writing them down?

Think of the segment of the real line from 3 to 19. It is $19 -3 = 16$ units long. If you walk along this line segment taking steps of length 4 units then you take $\large \frac{16}{4} \normalsize = 4$ steps, each footprint at the points of the sequence 3, 7, 11 and so on. Hence there are 4 steps after the footprint at 3 and hence there are $4 + 1 = 5$ footprints.

Now try the original question.

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