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第 14 週 · 等差與等比數列 · 第 3 節

等比數列:公比與通項

2、4、8、16… 每次乘同一個數 —— 這就是等比數列。等差是「加」的台階,等比是「乘」的暴漲。把公比 r 與通項 Tₙ = arⁿ⁻¹ 從零講透,配滿圖例。

▶ 第一關: 公比 r 與識別等比數列
24816x2x2x2T1=adoubles each step = r
248164÷2=28÷4=216÷8=2all ratios equal → geometric, r = 2
same start 2, 4 — then add vs multiplyT1T2T3T4+2 (arithmetic, straight)x2 (geometric, explodes)
▶ 第二關: 通項公式 Tₙ = arⁿ⁻¹
T1 = a(x r zero times)T2 = a r(x r once)T3 = a r²(x r twice)T4 = a r³(x r 3 times)Tn = a r^(n-1)x r is (n-1) times
Tn = a · rn-1first term acommon ratio rexponent = n-1only r carries the exponent — a stays as one factor
nTnexponential curvegeometric = exponential growth (W13)
▶ 第三關: r 的特殊情況與反求
r > 1: rises fast0 < r < 1: shrinksr < 0: alternates + - + -+-+
which term is 48 in 3, 6, 12, … ?3 x 2^(n-1) = 48 → 2^(n-1) = 16 = 2^4n-1 = 4 → n = 5match the exponent on the same base
is 96 in 3, 6, 12, … ?3 x 2^(n-1) = 96 → 2^(n-1) = 32 = 2^5 → n = 6integer → YESif exponent were not whole → NOT in sequence
⚔ BOSS: 給兩項求 r 與 a
T2 = 6, T4 = 54, r > 0① 2 steps apart: r² = 54 ÷ 6 = 9 → r = 3② back-substitute: a·r = 3a = 6 → a = 2