Question

在數(shù)列{an}中,若對(duì)任意的n均有an+an+1+an+2為定值(n∈N*),且a7=2,a9=3,a98=4,則此數(shù)列{an}的前100項(xiàng)的和S100=　　　　.

 

Answer 1

299

【解析】設(shè)定值為M,則an+an+1+an+2=M,進(jìn)而an+1+an+2+an+3=M,后式減去前式得an+3=an,即數(shù)列{an}是以3為周期的數(shù)列.由a7=2,可知a1=a4=a7=…=a100=2,共34項(xiàng),其和為68;由a9=3,可得a3=a6=…=a99=3,共33項(xiàng),其和為99;由a98=4,可得a2=a5=…=a98=4,共33項(xiàng),其和為132.故數(shù)列{an}的前100項(xiàng)的和S100=68+99+132=299.