Question

Dalam suatu barisan aritmetika, perbandingan jumlah 5 suku pertama dan jumlah 10 suku pertama adalah 2 : 3. Jika Un menyatakan suku ke n, maka nilai log (U5/U10 - 4 U10/U5 ) = ....
A. log 117 – log 11
B. log 3 – log 32
C. log 32 – log 34
D. log 7 – log 6
E. log 6 – log 7
UM UGM 2015
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Terima kasih

Answer 1

S5/S10 = 2/3

Sn = n/2 (2a + (n - 1)b)
S5 = 5/2 (2a + 4b)
= 5(a + 2b)
= 5a + 10b

S10 = 10/2 (2a + 9b)
= 5(2a + 9b)
= 10a + 45b

(5a + 10b)/(10a + 45b) = 2/3
5(a + 2b)/(5(2a + 9b)) = 2/3
(a + 2b)/(2a + 9b) = 2/3
3(a + 2b) = 2(2a + 9b)
3a + 6b = 4a + 18b
4a - 3a = 6b - 18b
a = -12b

U5 = a + 4b
= -12b + 4b
= -8b

U10 = a + 9b
= -12b + 9b
= -3b

log(u5/U10 - 4 U10/U5) = log(-8b/-3b - 4(-3b/-8b))
= log(8/3 - 3/2)
= log((16 - 9)/6)
= log(7/6)
= log7 - log6

Answer 2