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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.
How would you know if a function is linear or non-linear just by looking at a input-output table?
I don't know that you can do it just by looking at one. (At least, I'm not clever enough with arithmetic to do that.) But it's possible to do it using some simple aritmetic. Here's an input-output table.5 386 457 528 59The first thing I notice is that the numbers in the left-hand column are evenly spaced; the difference between any two of them is just one. The differences between all of the numbers in the right-hand column are also all the same, seven. So this input-output table represents a linear function.In case you're working in a slightly more advanced situation here's another example:3 117 239 2913 41In this case the left-hand column numbers are not evenly spaced and I can't just look at the differences between the numbers on the left. However, there's a slightly more advanced technique that I can apply.( 23 - 11 ) / ( 7 - 3 ) = 12 / 4 = 3( 29 - 23 ) / ( 9 - 7 ) = 6 / 2 = 3( 41 - 29 ) / ( 13 - 9 ) = 12 / 4 = 3The three slopes are the same. Therefore, the input-output table represents a linear function.
