Reciprocals
Problem
Prove that for any positive numbers $x_1$, $x_2$,..., $x_n$
Getting Started
These diagrams give hints for two alternative methods.
Student Solutions
This is another solution by Yatir from Maccabim-Reut High School, Israel.
Another challenge is to use the hints given by the two illustrations in the question and to give alternative proofs that the sum of a positive number and its reciprocal is greater than or equal to 2.
Yatir uses this inequality when he sums k fractions and their reciprocals in the following proof. Can you use a similar method to give a shorter proof of the result without resorting to mathematical induction? You will need to expand the expression given in (1), collect pairs of terms, decide how many pairs there are and use the inequality for each of the pairs of terms.