(+50) KuAT – Kuis Awal Tahun
Diketahui:
f(x) + f(x – 1) = x²
f(9) = 29
Ditanyakan:
f(63) + 23 = ...
Diketahui:
f(x) + f(x – 1) = x²
f(9) = 29
Ditanyakan:
f(63) + 23 = ...
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Verified answer
Jawaban:
2023
Penjelasan dengan langkah-langkah:
f(x) + f(x – 1) = x²
f(9) = 29
Ditanyakan:
f(63) + 23 = ...
f(x) = x²-f(x-1)
untuk x>9
untuk fungsi berikutnya kita tentukan fungsi genap dan ganjil
f(10) = 10²-f(9)
f(11) = 11²-f(11-1)
= 11²-f(10)
= 11²-(10²-f(9))
= 11²-10²+f(9)
f(12) = 12²-f(12-1)
= 12²-f(11)
= 12²-(11²-10²-f(9)
= 12²-11²+10²+f(9)
f(13) = 13²-f(13-1)
= 13²-f(12)
= 13²-(12²-11²+10²+f(9))
= 13²-12²+11²-10²-f(9)
maka diperoleh pola
untuk x ganjil
f(x) = x²-(x-1)²+(x-2)²-(x-3)²+.....+11²-10²+f(9)
f(63) = 63²-62²+61²-60²-....+11²-10²+29
= (63²-62²)+(61²-60²)+.....(11²-10²)+29
= (125)+(121)+(117)+.....+(21)+29
diperoleh barisan aritmatika dengan
a = 125
b = -4
n = (125-21)/4)+1
= (26)+1
= 27
Sn27 = 27/2 x (125(2)+(27-1)-4)
= 27/2 x (250-104)
= 27/2 x 146
= 1971
f(63) = 1971+29
= 2000
f(63)+23
= 2000+23
= 2023
f(x) = ½x² + ½x = ½x(x + 1).
Namun, karena f(9) = ½·9·10 = 45 ≠ 29, dan 45 = 29 + 16, maka pada f(x) terdapat suku ekstra yaitu 16·(–1)^x.
Sehingga, untuk x ganjil: f(x) = ½x(x + 1) – 16.
Untuk x genap: f(x) = ½x(x + 1) + 16.
f(63) + 23 = ½·63·64 – 16 + 23
= 32·63 – 16 + 23
= 16·2·63 – 16 + 23
= 16(126 – 1) + 23
= 16·125 + 23
= 2000 + 23
= 2023.
Penjelasan dengan langkah-langkah:
f(x) + f(x - 1) = x^2
f(x) = x^2 - f(x - 1)
f(9) = 29
Jika x > 9, maka x adalah bilangan bulat :
f(x) = x^2 - f(x - 1)
f(10) = 10^2 - f(10 - 1)
f(10) = 10^2 - f(9)
f(11) = 11^2 - f(11 - 1)
f(11) = 11^2 - f(10)
f(11) = 11^2 - (10^2 - f(9))
f(12) = 12^2 - f(12 - 1)
f(12) = 12^2 - f(11)
f(12) = 12^2 - (11^2 - (10^2 - f(9))
f(13) = 13^2 - f(13 - 1)
f(13) = 13^2 - f(12)
f(13) = 13^2 - (12^2 - (11^2 - (10^2 - f(9))))
Sehingg nilai f(63) :
f(63) = 63^2 - 62^2 + 61^2 - 60^2 + ..... + 11^2 - 10^2 + f(9)
f(63) = (63^2 - 62^2) + (61^2 - 60^2) + .... + (11^2 - 10^2) + f(9)
f(63) = (63 - 62)(63 + 62) + (61 - 60)(61 + 60) + .... + (11 - 10)(11 + 10) + f(9)
f(63) = (63 + 62 + 61 + 60 + .... + 11 + 10) + f(9)
Nilai dari f(63) + 23 :
f(63) + 23
= 1/2 x(x + 1) - 16 + 23
= 1/2 63(63 + 1) - 16 + 23
= 1/2 63(64) + 7
= 4.032/2 + 7
= 2.016 + 7
= 2.023