已知等差数列An的首项为a1公差为d都是实数,满足S2*S4/2+S3^2/9+2=0,求d的范围

Question

已知等差数列An的首项为a1公差为d都是实数,满足S2*S4/2+S3^2/9+2=0,求d的范围

Answer 1

由题意等差数列{aⁿ}的前n项的和Sn= na1 + n(n – 1)d/2,所以S2 = 2a1 + d,S3= 3a1 + 3d,S4 = 4a1 + 6d,代入原式可得(2a1+ d)(4a1 + 6d)/2 + (3a1 + 3d)2/9 + 2 = 0,所以(2a1+ d)(2a1 + 3d) + (a1 + d)2 + 2 = 0,所以4a12+ 8a1d + 3d2 + a12 + 2a1d+ d2 + 2 = 0,整理得5a12 + 10a1d + 4d2+ 2 = 0,根的判别式Δ= (10d)2– 4*5*(4d2 + 2) = 100d2 – 20(4d2 + 2) = 20d2– 40 ≥ 0,所以20d2≥ 40,因此d2 ≥ 2,可得d ≤ -√2,或者d ≥ √2 ；
综上所述,实数d的取值范围是(-∞,-√2]∪[√2,+∞) .