Известно, что число а = 1+корень5 / 2 называют з...

질문

Известно, что число а = 1+корень5 / 2
называют золотым числом, и оно является корнем уравнения х - х - 1 = 0. Найдите значение выражения а10 -55а -14
B
численном виде.

사고

Translating and interpreting math expression

I'm trying to decode the math problem in Russian. It mentions the golden ratio, φ = (1+√5)/2, which is a root of the equation x² - x - 1 = 0. The question asks for the numeric value of the expression a^10 - 55a - 14. There’s a case of possible typo, and I’ll use known Fibonacci identities to simplify the powers of φ.

Simplifying the expression

Okay, so using the Fibonacci identity for φ^10 = 55φ + 34, when I substitute into the expression a^{10} - 55a - 14, it simplifies nicely to 20. I double-checked with the exact numeric computations and confirmed the result is accurate: φ^10 = 122.991869... and the numeric value comes out cleanly as 20. Looks good!

Providing a brief explanation

I’ll calculate the result step-by-step while avoiding too much reasoning in detail. The identity φ^n = F_n φ + F_{n-1} helps simplify the expression, and the final numeric result comes out to 20. I'll proceed with this answer in Russian, correcting the typo in the equation to x² - x - 1 = 0.

Thought for 18s

답변

Ответ: 20.

Коротко почему: $\varphi=a=\frac{1+\sqrt5}{2}$ — корень $x^2-x-1=0$ , значит $a^2=a+1$ . Отсюда по стандартной формуле

a^{n}=F_n\,a+F_{n-1},

где $F_n$ — числа Фибоначчи. Для $n=10$ : $F_{10}=55,\;F_9=34$ , значит

a^{10}=55a+34.

Тогда

a^{10}-55a-14=(55a+34)-55a-14=20.

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사고

Translating and interpreting math expression

Simplifying the expression

Providing a brief explanation

답변

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