面积为6的直角三角形的长由小到大成等差数列,公差为d,求d的值.以最短边的长为首项,公差为d的等差数列中,102为第几项?

Question

面积为6的直角三角形的长由小到大成等差数列,公差为d,求d的值.
以最短边的长为首项,公差为d的等差数列中,102为第几项?

Answer 1

设首项为a1,则
a2=a1+d,a3=a1+2d
a1,a2,a3为直角三角形三边,则
a1²+a2²=a3²,a1²+a1²+2a1d+d²=a1²+4a1d+4d²,
a1²-2a1d-3d²=0,(a1-3d)(a1+d)=0,a1=3d
S=0.5*a1*a2=6,a1*a2=a1²+a1d=9d²+3d²=12,d=1,a1=3
则三角形三边为3,4,5
an=3+n-1=n+2
a102=104
答：d为1,102项为104