Algebra · Sequences · Recurrences · Inequalities · Fibonacci
The infinite sequence of natural numbers \(a_1, a_2, a_3, \dots\) is defined by
\[ a_1 = a_2 = 1, \qquad a_{n+2} = a_{n+1} + a_n \ \text{ for every } n \in \mathbb{N}. \]
Prove that
\[ \frac{a_1}{2} + \frac{a_2}{2^2} + \dots + \frac{a_{2024}}{2^{2024}} < 2. \]
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Serbian Municipal Competition 2024, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source