Search problems

2 problems match “harmonic numbers”. Clear

Visitors search a sample. Sign in for free to search every problem. Sign in

1. Harmonic numbers ●○○○○ 10 ptsCall three numbers $a, b, c$ harmonic when their reciprocals form an arithmetic sequence, that is, $\frac{1}{a}-\frac{1}{b}=\frac{1}{b}-\frac{1}{c}$. For example, $15, 12, 10$ are harmonic because $\frac{1}{12}-\frac{1}{15}=\frac{1}{10}-\frac{1}{12}$. Given that $x$, $5$, $3$ are harmonic numbers in this order with $x>5$, find $x$.
2. Ratio of weighted harmonic sums ●●●○○ 50 ptsDetermine the value of $\frac{\frac{2016}{1} + \frac{2015}{2} + \frac{2014}{3} + \dots + \frac{1}{2016}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2017}}$.