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# When do you use a variable in n expression instead of a number?

1) When a quantity (variable n) in an equation is not know and you are trying to evaluate it. For example, if a banana costs 5 cents, how many can you buy for 25 cents. We solve this by saying

Let n be the number of bananas that we can buy for 25 cents, then we have

5 x n = 25 and hence n = (25/5) = 5

2) When you want to express the general term in a series. For example the arithmetic series 1, 4, 7, 10, 13, 16,.......... is really

1, (1+3), (1+6), (1+9), (1+12), (1+15).......... and in general for such an arithmetic series you can find the 100th, 223rd, 567th term without writing down the long chain of numbers. How? we see a pattern in the arithmetic series. If we represent the starting number (1 in our case) as "a"and the equal increment from one term to the next (that's how an aritmetic series is defined) as "r" then the formlua "a + (n-1)r" gives you the n-th number. Remember that in the illustration above 1 is considered the 1st term in the series, (1+3) is the2nd term, (1+6) is the3rd term and so on.

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