2025 SSMO Accuracy Round Problems/Problem 1

Problem

An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies $a_n = a_1+a_2+\dots+a_{n-1}$ for all positive integers $n > 1$ . Given that $a_{20}=25,$ find $a_{25}$ .

Solution

Note that for every positive integer $n > 2$ , by the given recurrence relation, we have \begin{align*} a_n &= a_{n-1} + a_{n-2} + \cdots + a_1 \\ &= (a_{n-2} + \cdots + a_1) + a_{n-2} + \cdots + a_1 \\ &= 2(a_{n-2} + \cdots + a_1) \\ &= 2a_{n-1}. \end{align*} Therefore, $a_{25} = 2a_{24} = 4a_{23} = 8a_{22} = 16a_{21} = 32a_{20} = \boxed{800}$ .

~Sedro

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2025 SSMO Accuracy Round Problems/Problem 1

Problem

Solution