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periodic sequence
terms repeat in a cycle for k order (times) (e.g. -3,1,-3,1,-3… is periodic with order 2)
arithmetic series
Sn = n/2[2a + (n-1)d]
Sn = n/2(a+l)
arithmetic series proof
write out normal sum Sn= a + (a+d) + (a+2d)+…+(a+(n-2)d)+(a+(n-1)d), [1], and reverse, [2]
add [1] and [2]
geometric sequences
Un = ar^(n-1)
geometric series proof
Sn = a + ar + ar² + … + ar^(n-1)
multiply sum by r, rSn = ar + ar² + ar³ + … + ar^n
subtract [1] and [2], Sn - rSn = a-ar^n
Hence Sn(1-r) = a(1-r^n) and Sn = (a(1-r^n))/1-r
convergent geometric sequence condition
|r| < 1