Question

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

Answer 1

299

設(shè)定值為M,則a_n+a_n+1+a_n+2=M,進(jìn)而a_n+1+a_n+2+a_n+3=M,后式減去前式得a_n+3=a_n,即數(shù)列{a_n}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{a_n}的前100項(xiàng)的和S₁₀₀=68+99+132=299.