Using a(1 + r + r²) = 26 and a²(1 + r² + r⁴) = 364, dividing squares gives (1 + r + r²) / (1 – r + r²) = 13/7, leading to r = 3 or 1/3. The ...
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Writing x = ar³, y = ar⁹ and z = ar¹⁵, we compute y² = (ar⁹)² = a²r¹⁸. Also x x z = (ar³) x (ar¹⁵) = a²r¹⁸. Since y² = x x z, x, y, z form a GP.
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, ...
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.