Learning about Arithmetic and Geometric Sequences

So what’s the difference between arithmetic and geometric sequences?

Well, a sequence is where you have n “terms,” where for example 1 through 10. So if you have y=5x, a sequence for this from n_1 to n_{10} for would be {5, 10, 15, 20, 25, 30, 35, 40, 45, 10}. As you can tell, in a series where the variable such as x is multiplied by a constant, that results in there being a difference of that constant added toeach successive member of the sequence, so in the sequence above, there is a difference of 5 added between each term. This is linear; the sequence forms a straight line if graphed; by the y=mx + b form of a line, where m is the slope, here the difference of 5 between each term is also the slope, for each 1 unit change in x the independent variable, there is a 1 unit change in y the dependent variable. Note that for arithmetic series and linear equations the degree of the variable is 1, i.e x^1. A way of writing this is a_n = a_1 + (n-1)d, where a_1 is there first term, and d is the difference between terms.

On the other hand, a geometric sequence is where you have an exponential function, such as 2^x. Here, from n_1 to n_{10} would be {2, 4, 8, 16, 32, 64, 128, 256, 512, 1024}. Notice that here, between each successive term of the sequence, the difference continually increases, and each successive term is multiplied by 2. A way of writing this is gr^{n-1}, r is the multiple, and g is the multiple between each term and g is the first term in the sequence. So for the sequence above starting with 2, this would be 2*2^0 for n=1, which equals 2, 2*2^1 for n=2, which equals 4, etc.

Something interesting related is the distinction between linear functions, non-linear polynomial functions, and exponential functions. Linear is a polynomial where the degree is equal to one, like y=mx+b. A nonlinear polynomial is a polynomial where the degree is higher than one, such as y=mx^c+b where c is a constant and c>1. And an exponential function is where the variable is an exponent, such as y=m^x+b.

I think this is all correct, it might not be, I’m just reviewing this stuff myself really.

Tag: math

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