Mathematics OpenStudy (anonymous):

Let $x _{i}>0$ for i=1,2,3,...,n. For each positive integer k, prove that: $(x _{i}^{k}+...+x _{n}^{k})/n≤(x _{i}^{k+1}+...x _{n}^{k+1})/(x_{1}+x_{2}+...+x_{n})$ OpenStudy (anonymous):

Rewrite the equation as $\frac{x_1+\cdots+x_n}{n} \le \frac{x_1^{k+1}+\cdots +x_n^{k+1}}{x_1^k +\cdots + x_n^k}$ Note that this is the same as $\frac{x_1^1+\cdots + x_n^1}{x_1^0+\cdots +x_n^0}\le \frac{x_1^{k+1}+\cdots +x_n^{k+1}}{x_1^k +\cdots + x_n^k}$ There is a type of mean known as the Lehmer mean, which is $L_a=\frac{x_1^a+\cdots +x^a_n}{x^{a-1}_1+\cdots x^{a-1}_n}$ Using a bit of calculus, namely taking the derivative with respect to a, you'll known that L_a is a monotone increasing function with respect to a. Because it is increasing, you have that $p \le q \implies L_p \le L_q$ Note that your inequality is the same as $L_1 \le L_k$ which (using Lehmer's inequality) is obviously true for all k greater than or equal to one, which encompasses all positive integers. cuzican: Since the Constitution placed the sole power of impeachment in two political bodies, it is qualified as a political question. rthrth: Your gross income is $4,520.00/month. Your deductions are FICA (7.65%), federal tax withholding (11. 6 hours ago 1 Reply 0 Medals bonnie: Which lines in this excerpt from act V of Shakespeare's Romeo and Juliet create d 9 hours ago 1 Reply 0 Medals jeovonniwells21: Of the four sections of Justinian's Code, Institutiones was meant for: 5 hours ago 2 Replies 0 Medals emilee234: wrote somethin... lemme know what you guys think Its called Why? its not a poem or anything. 9 hours ago 12 Replies 0 Medals Sailor: Stuff i madeu2026 just gimme a second for the stuff to post in chat alright? 1 hour ago 10 Replies 4 Medals 81828: i need help Which function has an inverse that is also a function? g(x) = 2x u201 9 hours ago 1 Reply 2 Medals Olive2006: Help Please! 9 hours ago 2 Replies 0 Medals djmatthies12: Choose the correct sum of the polynomials (6x3 u2212 8x u2212 5) + (3x3 + 6x + 2) 11 hours ago 2 Replies 1 Medal djmatthies12: Zoya has to earn at least$300 to meet her fundraising goal. She has only 100 bracelets that she plans to sell at \$5 each.