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Geometric progression. Formula for the nth term and the sum of the first n terms

Definition
A geometric progression is a sequence where each next term is obtained by multiplying by the same number q0q\neq 0 (the common ratio). Recursively: bn+1=bnqb_{n+1}=b_n\cdot q, b10b_1\neq 0. If q>1|q|>1 the absolute values often grow; if 0<q<10<|q|<1 they tend to zero. The sign alternates if q<0q<0.
The ratio of neighbours is constant: bn+1bn=q\dfrac{b_{n+1}}{b_n}=q (where the denominator is nonzero)
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