$\begin{array}{ll}\\ 17.\quad&\textrm{Perhatikan hubungan berikut}\\ &\qquad\qquad\qquad\begin{aligned}1=&1^{\displaystyle 2}\\ 2+3+4=&3^{\displaystyle 2}\\ 3+4+5+6+7=&5^{2}\\ 4+5+6+7+8+9+10=&7^{\displaystyle 2}\\ \vdots\qquad &\vdots \end{aligned}\\ &\textrm{Tuliskan pola yang ada pada hubungan di atas dan buktikan} \end{array}$.
Lihat Bukti
Bukti: $\begin{aligned}&\textrm{Pola Umum}\\ &\:\:\textrm{Baris ke}-n\:(n=1,2,3,4,...)\\ &\circ \quad \textrm{ruas kanan berupa}:(2n-1)^{\displaystyle 2}\\ &\circ \quad \textrm{ruas kiri berupa penjumlahan dari}\:\:(2n-1)\:\: \textrm{bilangan bulat berurutan}\\ &\qquad\textrm{yang dimulai dari}\:\:n\\ &\qquad n+(n+1)+(n+2)+\cdots +(3n-2)=(2n-1)^{\displaystyle 2}\\ &\qquad \text{Catatan}:U_{\displaystyle n}=n+\: (2n-1-1).1=3n-2\\ &\qquad \textrm{Bentuk umum}:\sum_{_{\displaystyle k=0}}^{^{\displaystyle 2n-2}}(n+k)=(2n-1)^{\displaystyle 2}\\ &\qquad \bullet \quad k=0\longrightarrow U_{\displaystyle 1}=n+0=n\\ &\qquad \bullet \quad k=1\longrightarrow U_{\displaystyle 2}=n+1\\ &\qquad \bullet \quad k=2\longrightarrow U_{\displaystyle 3}=n+2\\ &\qquad \bullet \quad k=3\longrightarrow U_{\displaystyle 4}=n+3\\ &\qquad\qquad \vdots \\&\qquad \bullet \quad k=2n-2\longrightarrow U_{\displaystyle n}=n+(2n-2)=3n-2\\ &\textrm{Pembuktian Pola}\\ &\quad \circ \quad \textrm{Suku pertama}=a=n\\ &\quad \circ \quad \textrm{Beda}=b=1\\ &\quad \circ \quad \textrm{Banyak suku}=2n-1\\ &\quad \circ \quad \textrm{Suku terakhir}=3n-2\\ &\quad \circ \quad \textrm{Rumus Jumlahderet aritmetika }=\displaystyle \frac{2n-1}{2}\left( a+U_{\displaystyle n} \right)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:\:\: =\displaystyle \frac{2n-1}{2}\left( n+3n-2 \right)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:\:\: =(2n-1)^{\displaystyle 2}\\\end{aligned}$.
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Visualisasi Geometris Pola Matematika
Visualisasi Geometris $(2n-1)^2$
3 + 4 + 5 + 6 + 7 = 5² = 25
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