114 高雄市聯招
版主: thepiano
Re: 114 高雄市聯招
第 13 題
寫出幾項 a_n，可猜到 2^(n + 2) - 7(a_n)^2 = [2a_(n + 1) + a_n]^2
然後用數學歸納法證明
(1) n = 1 時，2^3 - 7(a_1)^2 = (2a_2 + a_1)^2 = 1
(2) 設 n = k 時，2^(k + 2) - 7(a_k)^2 = [2a_(k + 1) + a_k]^2
(3) n = k + 1 時
2^(k + 3) - 7[a_(k + 1)]^2
= 2{7(a_k)^2 + [2a_(k + 1) + a_k]^2} - 7[a_(k + 1)]^2
= [a_(k + 1)]^2 + 8[a_(k + 1)](a_k) + 16(a_k)^2
= [a_(k + 1) + 4(a_k)]^2
= [-2a_(k + 1) - 4(a_k) + a_(k + 1)]^2
= [2a_(k + 2) + a_(k + 1)]^2
寫出幾項 a_n，可猜到 2^(n + 2) - 7(a_n)^2 = [2a_(n + 1) + a_n]^2
然後用數學歸納法證明
(1) n = 1 時，2^3 - 7(a_1)^2 = (2a_2 + a_1)^2 = 1
(2) 設 n = k 時，2^(k + 2) - 7(a_k)^2 = [2a_(k + 1) + a_k]^2
(3) n = k + 1 時
2^(k + 3) - 7[a_(k + 1)]^2
= 2{7(a_k)^2 + [2a_(k + 1) + a_k]^2} - 7[a_(k + 1)]^2
= [a_(k + 1)]^2 + 8[a_(k + 1)](a_k) + 16(a_k)^2
= [a_(k + 1) + 4(a_k)]^2
= [-2a_(k + 1) - 4(a_k) + a_(k + 1)]^2
= [2a_(k + 2) + a_(k + 1)]^2