Sequences & Series
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relation between consecutive terms:
arithmetic sequence
geometric sequence
 arithmetic: difference
 geometric: ratio

to find n^{th} term of arithmetic sequence, need either:

to find n^{th} term of geometric sequence, need either:
 1 term & common ratio
 2 terms

arithmetic sequence:
general formula for n^{th} term
a_{n} = a_{1} + (n1)d

arithmetic sequence: recursive formula for n^{th} term
a_{n} = a_{n1} + d

formula to add first n terms of an arithmetic sequence if a_{1}, a_{n}, & n are known

formula to add first n terms of an arithmetic sequence if a_{n} unknown

geometric sequence: general formula for n^{th} term
a_{n} = a_{1} r^{ n1}

geometric sequence: recursive formula for n^{th} term
a_{n} = r a_{n1}

formula to add first
n terms of a geometric sequence as long as r
1

formula to add first
n terms of a geometric sequence when 1 < r < r & r
0

to make odd terms in a sequence negative, use factor:
(1)^{n}

to make even terms in a sequence negative, use factor:
(1)^{n+1}

how to determine ratio of a geometric sequence
 term multiplied by a constant equals the next term

when is the sum of all the terms in a geometric sequence equal to a number?
when common ratio is small: 1 < r < 1

difference between infinite sum formula & partial sum formula
the 1r^{n} in the numerator of the partial sum formula is replaced by 1 in the numerator of the infinite sum formula