Representing terms as a/r, a, ar gives product a³ = -1, so a = -1. Using sum -1/r – 1 – r = 13/12 gives common ratio r = -3/4 or -4/3, yielding terms 4/3, -1, 3/4 or 3/4, -1, ...
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With a = 2 and r = 4, explicit formula is tn = 2 x 4ⁿ⁻¹ and recursive formula is t1 = 2, tn = 4tn-1 for n >= 2. Solving 2 x 4ⁿ⁻¹ = 131072 gives 4ⁿ⁻¹ = 65536 ...
The sum formula is n(n + 1) / 2 > 1000, so n(n + 1) > 2000. Testing values gives 44 x 45 / 2 = 990 and 45 x 46 / 2 = 1035, so smallest n is 45.
Starting with 30 bacteria and doubling factor 2, the count after n hours is 30 x 2ⁿ. Thus, after 2 hours there are 120, after 4 hours 480 and after n hours 30 x 2ⁿ.
The sum of k consecutive natural numbers is k(2a + k – 1) / 2 = 100, meaning k(2a + k – 1) = 200. Testing odd factors gives only k = 5, yielding 18 + 19 + 20 + ...