Summation of a series

makada · 2007-09-21 02:08:42

This is a nice problem i got from a friend:

"There are 50 oranges which are laid out on a straight road in a linear fashion. The distance between the the first and second orange is one meter, between the second and third is three meters, between the third and fourth is five meters, ... and so on. A large collecting basket is placed next to the first orange. A man now has to collect all the oranges into the basket.
He starts next to the first orange and places it in the the basket. Then goes to the second orange and brings it to the basket . Then goes to the third orange and so on. At no point does the basket move and the man is not allowed to bring more than one orange at a time.
How many meters does the man cover eventually in collecting all 50 oranges?"

Now as I worked on the solution it seemed obvious the answer was a summation of a series of numbers.
The distance he traveled to get the first orange = 0 m (The basket is next to the first orange)
The distance he traveled to get the second orange = 2m (As he has to walk to and forth to the basket each time)
The distance he traveled to get the third orange = 8m
The distance he traveled to get the fourth orange= 18m
.
.
.
so on

The total distance he traveled for collecting all 50 oranges would be 0+2+8+18+......(total of 50 terms)
This is where I got stuck. I could not properly define the series in algebraic form and I could not find a formula for summation.
Any help?

luca-deltodesco · 2007-09-21 02:24:19

forget about distance there and back for now:

it goes:

1,3,5....(2n-1) from what you originally said

so the cumulate distance is:

so the total distance there and back for each apple is 2n^2

so you have:

Last edited by luca-deltodesco (2007-09-21 02:26:14)

mathsyperson · 2007-09-21 02:30:37

Almost, but you only need to sum up to 49.
That makes the answer 80850.

luca-deltodesco · 2007-09-21 02:53:02

ah yes ofcourse, i had removed the 0 entry from the beginning since it wasnt needed, and forgot about it at the end quite rightly so.

makada · 2007-09-21 08:42:52

That was nice.

On the same note, I want to know how to derive formulas for summation.
For Example: I know that the sum of first n positive integers is {n*(n+1)]/2. I also know how to prove it. But what I want to know how to derive this formula in the first place. Do I use integration or something else.

Same thing about sum of first n squares. How do I derive the formula in the first place?
I dont want the reverse proof , in which we already have the formula and go on to prove that it is correct.
Am I making any sense?

Is there any general rule for finding formulas for summation of a series? Or am i being too hopeful?

Last edited by makada (2007-09-21 08:44:15)

Math Is Fun Forum

#1 2007-09-21 02:08:42

Summation of a series

#2 2007-09-21 02:24:19

Re: Summation of a series

#3 2007-09-21 02:30:37

Re: Summation of a series

#4 2007-09-21 02:53:02

Re: Summation of a series

#5 2007-09-21 08:42:52

Re: Summation of a series

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