设等差数列An的前N项和为Sn,若S4大于等于10,S5小于等于15,则A4最大值为多少?

Question

设等差数列An的前N项和为Sn,若S4大于等于10,S5小于等于15,则A4最大值为多少?

Answer 1

∵等差数列{a_n}的前n项和为Sn,且S4≥10,S5≤15
∴ S4=4a1+ 4×32d≥10S5=5a1+ 5×42d≤15
即 2a1+3d≥5a1+2d≤3
∴ a4=a1+3d≥ 5-3d2+3d≥ 5+3d2a4=a1+3d=(a1+2d)+d≤3+d
∴ 5+3d2≤a4≤3+d,5+3d≤6+2d,d≤1
∴a4≤3+d≤3+1=4故a4的最大值为4,
故答案是4