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October 3rd, 2019, 10:21 PM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 477 Thanks: 1  Explain this step in Average speed problem?
What is the average speed of a man who works at 1kmph for 1 hour, 2 kmph for 2 hours........ till 10 kmph for 10 hours? A) Average speed = (Total distance)/(Total time) = (1*1+2*2+3*3…….10*10)/(1+2+3****+10) = {1/6 n(n+1)(2n+1)}/ {1/2 n(n+1)} = 2n+1/3; (n = 10) = 7 Explain this step in Average speed problem: {1/6 n(n+1)(2n+1)}/ {1/2 n(n+1)}? 
October 3rd, 2019, 10:51 PM  #2 
Senior Member Joined: Jun 2019 From: USA Posts: 310 Thanks: 162 
Those are tricks for quickly finding the sums 1²+2²+3²+...+n² and 1+2+3+...+n. The second one, you can visualise as making a "triangle" of a row of 1 block, a row of 2 blocks, a row of 3 blocks, a row of 4 blocks, etc. If you make a second triangle, turn it around, and put it next to the first, you get a rectangle that is n blocks by n+1 blocks. (There was also a possibly apocryphal story about some maths or physics genius doing this when his teacher tried to give him busy work to shut him up.) The first one is similar, but instead of a triangle, you're building a pyramid with a plane of 1x1 block, a plane of 2x2 blocks, a plane of 3x3 blocks, a plane of 4x4 blocks, etc. 
October 4th, 2019, 02:58 AM  #3 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 132 Thanks: 49 Math Focus: Area of Circle 
It is a weighted average $\dfrac{w_1a_1+w_2a_2+...w_na_n}{w_1+w_2+...w_n}$, sort of (in this case, what I call quadratic average $\dfrac{{a_1}^2+...{a_n}^2}{a_1+...a_n}$). $Total Distance =Distace_1 + Distance_2 + ... Distance_{10} = 1^2+2^2+...10^2=\dfrac{10 \times 11 \times 21}{6}=385$ $Total Time = Time_1+Time_2+...Time_{10}=1+2+...10=55$ $Average Speed=\dfrac{TotalDistance}{TotalTime}=385/55=7$ 
October 4th, 2019, 03:54 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,035 Thanks: 2271 
Any guesses as to what kind of work the man is doing?

October 4th, 2019, 04:31 AM  #5 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 132 Thanks: 49 Math Focus: Area of Circle  Selfindulging, while slow driving... Or fast walking...
Last edited by skipjack; October 4th, 2019 at 03:03 PM. 
October 4th, 2019, 07:29 AM  #6 
Senior Member Joined: Jun 2019 From: USA Posts: 310 Thanks: 162 
For 55 hours? Now that's what I call stamina.

October 4th, 2019, 08:35 AM  #7 
Senior Member Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 
The universe is light and full of stamina . 

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