Whakaoti i te 1/9!+1/10!+1/11!=122/11!

$$\frac { 1 } { 9 ! } + \frac { 1 } { 10 ! } + \frac { 1 } { 11 ! } = \frac { 122 } { 11 ! }$$

Manatoko

Ngā Upane Otinga

Ko te huarea o 9 ko 362880.

$$\frac{1}{362880}+\frac{1}{10!}+\frac{1}{11!}=\frac{122}{11!}$$

Ko te huarea o 10 ko 3628800.

$$\frac{1}{362880}+\frac{1}{3628800}+\frac{1}{11!}=\frac{122}{11!}$$

Ko te maha noa iti rawa atu o 362880 me 3628800 ko 3628800. Me tahuri \frac{1}{362880} me \frac{1}{3628800} ki te hautau me te tautūnga 3628800.

$$\frac{10}{3628800}+\frac{1}{3628800}+\frac{1}{11!}=\frac{122}{11!}$$

Tā te mea he rite te tauraro o \frac{10}{3628800} me \frac{1}{3628800}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{10+1}{3628800}+\frac{1}{11!}=\frac{122}{11!}$$

Tāpirihia te 10 ki te 1, ka 11.

$$\frac{11}{3628800}+\frac{1}{11!}=\frac{122}{11!}$$

Ko te huarea o 11 ko 39916800.

$$\frac{11}{3628800}+\frac{1}{39916800}=\frac{122}{11!}$$

Ko te maha noa iti rawa atu o 3628800 me 39916800 ko 39916800. Me tahuri \frac{11}{3628800} me \frac{1}{39916800} ki te hautau me te tautūnga 39916800.

$$\frac{121}{39916800}+\frac{1}{39916800}=\frac{122}{11!}$$

Tā te mea he rite te tauraro o \frac{121}{39916800} me \frac{1}{39916800}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{121+1}{39916800}=\frac{122}{11!}$$

Tāpirihia te 121 ki te 1, ka 122.

$$\frac{122}{39916800}=\frac{122}{11!}$$

Whakahekea te hautanga \frac{122}{39916800} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$\frac{61}{19958400}=\frac{122}{11!}$$

Ko te huarea o 11 ko 39916800.

$$\frac{61}{19958400}=\frac{122}{39916800}$$

Whakahekea te hautanga \frac{122}{39916800} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$\frac{61}{19958400}=\frac{61}{19958400}$$

Whakatauritea te \frac{61}{19958400} me te \frac{61}{19958400}.

$$\text{true}$$