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SMUP 2010

Solusi SMUP 2010 Kode 041 – Nomor 17


Misalkan\ N\ adalah\ himpunan\ semua\ bilangan\ bulat\ positif.\\Misalkan\ f: N \rightarrow N\ suatu\ fungsi\ sehingga\\f(x+1)=f(x)+x\\untuk\ semua\ x\in N\ dan\ f(1)=5.\\Nilai\ dari\ f(2010)\ adalah\ ....\\A.\quad  2.019.050\\B.\quad 1.019.500\\C.\quad 1.019.510\\D.\quad 3.019.050\\E.\quad 4.019.500
Jawaban: A

f(x)=f(x+1)-x
f(1)=f(2)-1=5\quad \rightarrow \quad f(2)=6
f(2)=f(3)-2=6\quad \rightarrow \quad f(3)=8
f(3)=f(4)-3=8\quad \rightarrow \quad f(4)=11
f(4)=f(5)-4=11\quad \rightarrow \quad f(5)=15
f(5)=f(6)-5=15\quad \rightarrow \quad f(6)=20

Dari\ pola\ tersebut\ diperoleh\ bahwa\\untuk\ x=n\quad \rightarrow \quad f(n)=f(1)+\big(1+2+3+...+(n-1)\big)
sehingga:

f(n)=5+\big(\frac{n-1}{2}\big)n
\begin{array}{rcl}f(2010)&=&5+\big(\frac{2009}{2}\big)\times 2010\\\\&=&5+2009\times 2010\\\\&=&2.019.050\end{array}

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