岡山大学
2010年 文系 第2問
2
![自然数m,nに対して,自然数m\diamondnを次のように定める.\begin{center}\setlength\unitlength{1truecm}\begin{picture}(12,5)(0,0)\put(0.5,0){\line(0,1){5}}\put(0,4.5){\line(1,0){5.5}}\put(0.12,4.7){\diamond}\put(0.12,3.9){1}\put(0.12,3.1){2}\put(0.12,2.3){3}\put(0.12,1.5){4}\put(0.12,0.7){5}\put(0.15,0){\vdots}\put(0.8,4.7){1}\put(0.8,3.9){4}\put(0.8,3.1){9}\put(0.7,2.3){16}\put(0.7,1.5){25}\put(0.7,0.7){36}\put(0.85,0){\vdots}\put(1.6,4.7){2}\put(1.6,3.9){6}\put(1.5,3.1){13}\put(1.5,2.3){22}\put(1.5,1.5){33}\put(1.5,0.7){46}\put(1.65,0){\vdots}\put(2.4,4.7){3}\put(2.4,3.9){8}\put(2.3,3.1){17}\put(2.3,2.3){28}\put(2.3,1.5){41}\put(2.3,0.7){56}\put(2.45,0){\vdots}\put(3.2,4.7){4}\put(3.1,3.9){10}\put(3.1,3.1){21}\put(3.1,2.3){34}\put(3.1,1.5){49}\put(3.1,0.7){66}\put(3.25,0){\vdots}\put(4,4.7){5}\put(3.9,3.9){12}\put(3.9,3.1){25}\put(3.9,2.3){40}\put(3.9,1.5){57}\put(3.9,0.7){76}\put(4.05,0){\vdots}\put(4.8,4.7){・・・}\put(4.8,3.9){・・・}\put(4.8,3.1){・・・}\put(4.8,2.3){・・・}\put(4.8,1.5){・・・}\put(4.8,0.7){・・・}\put(4.8,0){\ddots}\put(7.5,1){\line(0,1){3}}\put(7,3.5){\line(1,0){3.5}}\put(7.1,3.7){\diamond}\put(7,2.3){m}\put(8.8,3.7){n}\put(8.5,2.3){m\diamondn}\end{picture}\end{center}例えば,1\diamond1=4,1\diamond2=6,2\diamond1=9,4\diamond2=33,5\diamond3=56,1\diamond6=14,6\diamond1=49である.(1)数列8\diamond1,8\diamond2,8\diamond3,・・・の初項8\diamond1から第25項8\diamond25までの和を求めよ.(2)m\diamondn=474を満たす自然数m,nの組をすべて求めよ.](./thumb/612/1190/2010_2.png)
2
自然数$m,\ n$に対して,自然数$m \diamond n$を次のように定める.
\begin{center}
\setlength\unitlength{1truecm}
\begin{picture}(12,5)(0,0)
\put(0.5,0){\line(0,1){5}}
\put(0,4.5){\line(1,0){5.5}}
\put(0.12,4.7){$\diamond$}
\put(0.12,3.9){1}
\put(0.12,3.1){2}
\put(0.12,2.3){3}
\put(0.12,1.5){4}
\put(0.12,0.7){5}
\put(0.15,0){$\vdots$}
\put(0.8,4.7){1}
\put(0.8,3.9){4}
\put(0.8,3.1){9}
\put(0.7,2.3){16}
\put(0.7,1.5){25}
\put(0.7,0.7){36}
\put(0.85,0){$\vdots$}
\put(1.6,4.7){2}
\put(1.6,3.9){6}
\put(1.5,3.1){13}
\put(1.5,2.3){22}
\put(1.5,1.5){33}
\put(1.5,0.7){46}
\put(1.65,0){$\vdots$}
\put(2.4,4.7){3}
\put(2.4,3.9){8}
\put(2.3,3.1){17}
\put(2.3,2.3){28}
\put(2.3,1.5){41}
\put(2.3,0.7){56}
\put(2.45,0){$\vdots$}
\put(3.2,4.7){4}
\put(3.1,3.9){10}
\put(3.1,3.1){21}
\put(3.1,2.3){34}
\put(3.1,1.5){49}
\put(3.1,0.7){66}
\put(3.25,0){$\vdots$}
\put(4,4.7){5}
\put(3.9,3.9){12}
\put(3.9,3.1){25}
\put(3.9,2.3){40}
\put(3.9,1.5){57}
\put(3.9,0.7){76}
\put(4.05,0){$\vdots$}
\put(4.8,4.7){$\cdots$}
\put(4.8,3.9){$\cdots$}
\put(4.8,3.1){$\cdots$}
\put(4.8,2.3){$\cdots$}
\put(4.8,1.5){$\cdots$}
\put(4.8,0.7){$\cdots$}
\put(4.8,0){$\ddots$}
\put(7.5,1){\line(0,1){3}}
\put(7,3.5){\line(1,0){3.5}}
\put(7.1,3.7){$\diamond$}
\put(7,2.3){$m$}
\put(8.8,3.7){$n$}
\put(8.5,2.3){$m \diamond n$}
\end{picture}
\end{center}
例えば,$1 \diamond 1=4,\ 1 \diamond 2=6,\ 2 \diamond 1=9,\ 4 \diamond 2=33,\ 5 \diamond 3=56,\ 1 \diamond 6=14,\ 6 \diamond 1=49$である.
(1) 数列$8 \diamond 1,\ 8 \diamond 2,\ 8 \diamond 3,\ \cdots$の初項$8 \diamond 1$から第25項$8 \diamond 25$までの和を求めよ.
(2) $m \diamond n=474$を満たす自然数$m,\ n$の組をすべて求めよ.
(1) 数列$8 \diamond 1,\ 8 \diamond 2,\ 8 \diamond 3,\ \cdots$の初項$8 \diamond 1$から第25項$8 \diamond 25$までの和を求めよ.
(2) $m \diamond n=474$を満たす自然数$m,\ n$の組をすべて求めよ.
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