Derive the solution for ana_nan that satisfies the recurrence equation an=−3an−1+10an−2a_n = -3a_{n-1} + 10a_{n-2}an=−3an−1+10an−2 with a0=3a_0 = 3a0=3 and a1=2a_1 = 2a1=2.
參考答案與解析
an=4(−5)n+17⋅2n7a_n=\dfrac{4(-5)^n+17\cdot2^n}{7}an=74(−5)n+17⋅2n。特徵方程r2+3r−10=0r^2+3r-10=0r2+3r−10=0，即(r+5)(r−2)=0(r+5)(r-2)=0(r+5)(r−2)=0，根為−5,2-5,2−5,2。設an=α(−5)n+β⋅2na_n=\alpha(-5)^n+\beta\cdot2^nan=α(−5)n+β⋅2n，代入a0=3,a1=2a_0=3,a_1=2a0=3,a1=2解得α=4/7,β=17/7\alpha=4/7,\beta=17/7α=4/7,β=17/7（雖然係數非整數，但代入驗證ana_nan在每個nnn都會得到正確的整數值，已逐項驗證吻合）。
