Math 11 Sequences and Series Blog Post

 

Arithmetic
General Term: t_{n} = a + (n-1)d
To figure out the value of any given term (t_{n}), a (the first term), d (the common difference), or n (the term in which a certain value appears).
Sum of Terms: s_{n} = [(a+t_{n})n]\div2
To figure out the sum of any given arithmetic series (S_{n}), n (the last term), or a (the first term).

Geometric
General Term: t_{n} = ar^{n-1}
To figure out any given term (t_{n}), a (the first term), r (the common ratio), or n (the term in which a certain value appears).
Sum of Terms: S_{n} = [a(r^n-1)]\div (r-1)
To figure out the partial sum of any geometric series if not given the last term, the last term, the first term, or the common ratio OR
S_{n} = (rt_{n} - a)\div (r-1)
to do the same thing, but if given the last term of the series.

You can have two types of geometric series: convergent (terms come closer together) or divergent (terms become further apart).
When you have a convergent series, the common ratio is greater than -1 but smaller than 1 (and cannot equal zero). Convergent series have infinite sums (the number in which the sum of all the terms converge at), which can be calculated with the formula S_{\infty} = a\div (1-r).
Divergent series don’t have infinite sums.

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