设{a}是公差为正整数的等差数列,若a1+a2+a3=15,a1*a2*a3=80,求a11+a12+a13
问题描述：
设{a}是公差为正整数的等差数列,若a1+a2+a3=15,a1*a2*a3=80,求a11+a12+a13
答
设公差为d a1=a2-d a3=a2+d
a1+a2+a3=15 => a2=5
a1*a2*a3=a2*(a2^2-d^2)=80 => d^2=9 d=3(d>0)
a11+a12+a13=3a12=3a2+30d=105
答
a1+a3=2a2
=>3a2=15
=>a2=5
a+d=5①
a1*a2*a3=80
a1^2*a2(1+2d)=80
=>a1^2*(1+2d)=16②
联立①②
解得,a1=2,a3=8 d=3
a11+a12+a13
=3a12
=3(a1+11d)
=3*(2+33)
=105