範例4 指、對數形成的數列與級數 綜合 10. 已知a1,a2,ds,...,a是首項為-3,公差為一的等差數列,設b=2x2-3x23x2aux... x 2(-1)atlan ,試選出正確的選項。 (B)(D)(E) (A) log2 (b12) = 2 (B) log2 (b21) = 13 (D)log: (bz),logz (ba),log(b),log(b),...為等差數列 (E)log: (b),log2(b),log2(b),log(b),...為等差數列 an=q+(n-1)d=-3+ 2 (n-1)= n-10 3 , bn=29-a2+a3-94+...+(-1)+la =...=91-92=-d=-1 (A)b2=21-a2+a3-9q+..+G1-G2, 又aq-d2=93-dq=" (B) b21 = 21-a2+a3-a4+ ··· +a19-20 × 2921 = (2 log2(62) = log2. 1 4 =- 1×23 = 23, log2(621) 3 (C) bn+1 = 2-3 xb (C). ④提示 bn+2 = :2(-1)h+2an. <2(-1)+3d an+2 bn ·an+1. x (D(E)利用(C)的結果,檢查 log₂ (bn+2)-log(bn) (bn+2) = log2(- bn 1 1 1 1 1 所以b12 = = 2 3 3 3 3 3 字= 2-2 = 1 = ⇒ 4 21-10 10 =2言, 解 bn+2 = (C)(D)(E) =2(-1)+2an+1x2(-1)2+3 a Dan+2=2(-1)n+2(an+1-an+2) =2 (-1)+(-) bm bn 若n 為偶數,則 Da+2 = 25 ⇒ log:(benz)-log:(b)=log:(02)=log:(23)=-13 bn 若n為奇數,則 bn+2= 23 bn → log:(bntz)-logz(b)=log:(012)= bnx2)=log:(23) = 2 bn 3 故(D)是公差為 - 1 的等差數列,(E)是公差為 ㄒㄧㄣ 的等差數列