展開する (2^r(r^2-2))/((r+1)^2(r+2)^2)

解

展開する

\frac{2 ^{r} ( r ^{2} - 2 )}{( r + 1 ) ^{2} ( r + 2 ) ^{2}}

\frac{2 ^{r} r ^{2} - 2 ^{1 + r}}{r ^{4} + 6 r ^{3} + 13 r ^{2} + 12 r + 4}

解答ステップ

\frac{2 ^{r} ( r ^{2} - 2 )}{( r + 1 ) ^{2} ( r + 2 ) ^{2}}

(r + 1)^{2} = r^{2} + 2 r + 1

= \frac{2 ^{r} ( r ^{2} - 2 )}{( r ^{2} + 2 r + 1 ) ( r + 2 ) ^{2}}

(r + 2)^{2} = r^{2} + 4 r + 4

= \frac{2 ^{r} ( r ^{2} - 2 )}{( r ^{2} + 2 r + 1 ) ( r ^{2} + 4 r + 4 )}

拡張

(r^{2} + 2 r + 1) (r^{2} + 4 r + 4) : r^{4} + 6 r^{3} + 13 r^{2} + 12 r + 4

= \frac{2 ^{r} ( r ^{2} - 2 )}{r ^{4} + 6 r ^{3} + 13 r ^{2} + 12 r + 4}

拡張

2^{r} (r^{2} - 2) : 2^{r} r^{2} - 2^{1 + r}

= \frac{2 ^{r} r ^{2} - 2 ^{r + 1}}{r ^{4} + 6 r ^{3} + 13 r ^{2} + 12 r + 4}