Numerical answer
A sequence is defined by a₁ = 2 and aₙ₊₁ = 3aₙ − 2 for n ≥ 1. Find a₁ + a₂ + a₃ + a₄ + a₅.
Show answer and solution
Answer: 126
Rewrite: aₙ₊₁ − 1 = 3(aₙ − 1). So bₙ = aₙ − 1 is a GP with b₁ = 1 and ratio 3, i.e. aₙ = 1 + 3ⁿ⁻¹.
Terms: 2, 4, 10, 28, 82.
Sum = 5 + (1 + 3 + 9 + 27 + 81) = 5 + 121 = 126.
Tip: For aₙ₊₁ = p·aₙ + q, shift by the fixed point c = q/(1 − p) (here c = 1) to turn it into a pure GP.
Terms: 2, 4, 10, 28, 82.
Sum = 5 + (1 + 3 + 9 + 27 + 81) = 5 + 121 = 126.
Tip: For aₙ₊₁ = p·aₙ + q, shift by the fixed point c = q/(1 − p) (here c = 1) to turn it into a pure GP.
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