Pembahasan Soal Deret Geometri Takhingga Berganti Tanda

Soal:

Jumlah deret geometri takhingga berikut:

8 - 8/3 + 8/9 - 8/27 + . . . adalah:

A. 5 2/9

B. 6

C. 6 2/9

D. 12

E. 24

(Matematika P54 Utama, Ujian Nasional Tahun Pelajaran 2009/2010, 

SMA/MA Program Studi IPS/Keagamaan)

Pembahasan: 

Misalkan jumlah barisan di atas adalah:

         S_~  =  8 – 8/3 + 8/9 – 8/27 + 8/81 - . . .

= 8[1 – 1/3 + 1/9 – 1/27 + 1/81 - . . . ]

= 8[1 + 1/9 + 1/81+ . . .  - (1/3 + 1/27+ . . . ) ]

= 8[(1 + 1/9 + 1/81+ . . . ) - (1/3 + 1/27+ . . . ) ]

          A _~ = 1 + 1/9 + 1/81+ . . .                  B_~ = 1/3 + 1/27+ . . .

                             a = 1                                        a = 1/3

                             r = 1/9                                     r = 1/9

          A _~ = a/(1– r) = 1/(1 – 1/9)  = 1/(8/9) = 9/8                

          B _~ = a/(1– r) = 1/3/(1 – 1/9)  = (1/3)/(8/9) = 1/3 . 9/8 = 3/8

          S_~ = 8[A_~ – B_~] = 8[9/8 – 3/8] = 8(6/8) = 6.                   Jawab: B