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Home » Taking the square root of both sides, we find: - Carbonext

Taking the square root of both sides, we find: - Carbonext

📅 February 26, 2026 👤 scraface

Feb 26, 2026

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📌 Define \( L(u) = u - rac{u^3}{3} \) for every real number \( u \). If \( n \) is a positive integer, define \( a_n \) by \( a_1 = rac{1}{2} \), \( a_{n+1} = L(a_n) \). Find \( \lim_{n o \infty} a_n \).

📌 We are given a recursive sequence:

📌 \( a_1 = rac{1}{2} \)

📌 \( a_{n+1} = L(a_n) = a_n - rac{a_n^3}{3} \)

📌 We seek \( \lim_{n o \infty} a_n \), assuming the limit exists and equals \( L \).

📌 If the limit exists, then:

📌 L = L - rac{L^3}{3} \Rightarrow rac{L^3}{3} = 0 \Right

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R(9) &= 3(81) + 18 + 1 = 243 + 18 + 1 = 262, \\
R(10) &= 3(100) + 20 + 1 = 300 + 20 + 1 = 321.
We are to find \( \gcd(R(1), R(2), \ldots, R(10)) = \gcd(6, 17, 34, 57...
Notice:
\( R(2) = 17 \), a prime number.
Check whether 17 divides any of the other terms modulo 17:
R(1) & = 6 \mod 17 = 6, \\
R(3) & = 34 \equiv 0 \mod 17, \\
R(4) & = 57 \equiv 57 - 3\times17 = 57 - 51 = 6 \mod 17, \\
R(5) & = 86 \equiv 86 - 5\times17 = 86 - 85 = 1 \mod 17, \\

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