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Omni-MATH / For each positive integer n , let …0<a, b, c, d<1000000 such that T₁₉₈₈=aS₁₉₈₉−b and U₁₉₈₈=cS₁₉₈₉−d .

Problem

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problem

For each positive integer nn , let \begin{align*} S_n &= 1 + \frac 12 + \frac 13 + \cdots + \frac 1n \\ T_n &= S_1 + S_2 + S_3 + \cdots + S_n \\ U_n &= \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{align*} Find, with proof, integers 0<a,b,c,d<10000000 < a,\ b,\ c,\ d < 1000000 such that T1988=aS1989−bT_{1988} = a S_{1989} - b and U1988=cS1989−dU_{1988} = c S_{1989} - d .
Plain-text mathematical notation (without MathML)
For each positive integer  n , let \begin{align*} S_n &= 1 + \frac 12 + \frac 13 + \cdots + \frac 1n \\ T_n &= S_1 + S_2 + S_3 + \cdots + S_n \\ U_n &= \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{align*} Find, with proof, integers 0<a, b, c, d<1000000 such that T₁₉₈₈=aS₁₉₈₉−b and U₁₉₈₈=cS₁₉₈₉−d .
Original LaTeX notation
For each positive integer  $n$ , let \begin{align*} S_n &= 1 + \frac 12 + \frac 13 + \cdots + \frac 1n \\ T_n &= S_1 + S_2 + S_3 + \cdots + S_n \\ U_n &= \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{align*} Find, with proof, integers $0 < a,\ b,\ c,\ d < 1000000$ such that $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$ .

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Source and history

task_id
065bf0e8-0efa-51a0-ad50-ea62c4ff89a9
task_key
test--065bf0e8-0efa-51a0-ad50-ea62c4ff89a9
task_revision_id
1
config / split
/ test

Official source

initial import