1.6.2—Progressions

An arithmetic progression has constant difference d, with uₙ=a+(n−1)d and Sₙ=n/2[2a+(n−1)d]. A geometric progression has constant ratio r, with uₙ=arⁿ⁻¹ and Sₙ=a(rⁿ−1)/(r−1) when r≠1.

Check consecutive differences or ratios before choosing a formula. For an infinite geometric sum, convergence requires |r|<1 and S∞=a/(1−r).

For 3,6,12,…, a=3 and r=2, so it has no finite infinite sum because |r|>1.

A sequence can have a convergent partial-sum pattern without being geometric; never infer r from non-consecutive terms alone.