Difference between revisions of "AoPS Wiki talk:Problem of the Day/September 7, 2011"
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Raising that thing to the seventh doesn't look appealing, nor does solving for <math>x</math>. So we try getting to <math>x^7+\frac{1}{x^7}</math> gradually. | Raising that thing to the seventh doesn't look appealing, nor does solving for <math>x</math>. So we try getting to <math>x^7+\frac{1}{x^7}</math> gradually. | ||
− | Since <math>x+\frac{1}{x}=1</math>, <math>\left(x^2+\frac{1}{x}\right)^2=x^2+\frac{1}{x^2}=1</math> and thus <math>x^2+\frac{1}{x^2}=-1</math>. | + | Since <math>x+\frac{1}{x}=1</math>, <math>\left(x^2+\frac{1}{x}\right)^2=x^2+\frac{1}{x^2}+2=1</math> and thus <math>x^2+\frac{1}{x^2}=-1</math>. |
We can also compute: | We can also compute: |
Latest revision as of 10:16, 7 September 2011
Solution
Raising that thing to the seventh doesn't look appealing, nor does solving for . So we try getting to
gradually.
Since ,
and thus
.
We can also compute:
so
.
Squaring , we see
and thus
. Thus,
and therefore,
.