Question

Is there a way to calculate a minimum value of the expression

Answer 1

\[\huge \left| 2x-1 \right| +\left| 2x-2 \right| + \left| 2x-3 \right|..... + \left| 2x-81 \right|\]

Answer 2

@mathslover @ganeshie8

Answer 3

Is anyone here

Answer 4

820 for x= 41/2?

Answer 5

what?

Answer 6

I think you can do this by induction first think about 2 terms then 3...so on
like for the first two its 4x = 1+2 then for 3 its 4x = 1+2+3..

Answer 7

ahhh , I got it , but not by induction, uhh yes !!

Answer 8

how?

Answer 9

I think it would be same for any value of 2x between 1 and 81 because these are just distances and going one side will just make the other side bigger

Answer 10

I tries to think by logic:-
So suppose there are three friends

A , B ,C

A lives on the number line at -5

B lives on the number line at 6

C lives on the number line at 10


If they had to meet at a place so that minimum distance should be covered then
If you carefully observe the minimum distance would be when all of them meet at
B
which is 3+1/2
4/2
=2 2nd friend which = B
If you apply the same logic to the question

So ,Let's say there are 81 boxes (as the last term is 81) so minimum must lie on

81+1/2
= 41 st box

(41*2)-1 = 81
(41*2)-2= 80

81 , 80 ,79 ........1 ,0 , 81 ,80 ,79........1
Sum of the natural numbers =
n(n+1)/2

\[\frac{ 81(82) }{ 2 }*2 = 6642\]

Answer 11

tried*

Answer 12

@aviz

Answer 13

Wouldn't it only go till |2x-81| = 1?

Answer 14

yes it would that's what i wrote

Answer 15

if x= 41/2 then it will go from 40, 39, 38..1, 0 , 1 , 2, .....40 and the answer will be 40(41)/2 = 820

Answer 16

sorry *2 = 820*2 =1640

Answer 17

it won't be 41/2
minimum would lie on 41

Answer 18

http://www.wolframalpha.com/input/?i=%5Cleft%7C+2x-1+%5Cright%7C+%2B%5Cleft%7C+2x-2+%5Cright%7C+%2B+%5Cleft%7C+2x-3+%5Cright%7C%2B...+%2B+%5Cleft%7C+2x-81+%5Cright%7C

Answer 19

And thus x would be considered 41

Answer 20

@No.name ?

Answer 21

In the link , that does not give the minimum

Answer 22

look at the plot, the min value is between 1500 and 2000, occurs near 20
going thru ur method now :)

Answer 23

what @aviz

Answer 24

Minimum is 1640 at 20.5

Answer 25

@No.name : your method would have been correct if it had bee |x-a| form bu its |bx-a| form hence division of middle value by 2.. your logic is very good and correct but for this slight hitch

Answer 26

yes
http://www.wolframalpha.com/input/?i=%5Csum+%5Climits_%7Bk%3D1%7D%5E81+%7C2%2820.5%29-k%7C

Answer 27

Nice :)

Answer 28

Oh , I see , my bad

Answer 29

So ,41/2

Answer 30

Yes :)

Answer 31

nice work @ganeshie8 !

Answer 32

40 , 39 , 38 .....1 ,0 , 1, 2, 3 ,4 ,..... 49

\[\frac{ 40*41 }{ 2 }*2 = 1640\]

Answer 33

nice work @ganeshie8

Answer 34

But i am getting 1640 not anywhere close to even 20

Answer 35

is the last term |2x - 81|?

Answer 36

yes

Answer 37

The mod got cut-off

Answer 38

Well , this isn't a problem from any textbook , but from my own mind

Answer 39

So this isn't any official problem

Answer 40

choose the middle value ..

Answer 41

Yeah it is the value where the minimum will lie
The value is 41/2

Answer 42

note that as x-> infty ... the whole expression goes to infinity
and as x->-infty ... the whole expression goes to infinity.
so it must be symmetric about certain point. take your middle expression to be zero.
you will get the minimum value.

Answer 43

the x doent go to infinity

Answer 44

dosen't*

Answer 45

woops!! something went wrong in my calculation.

Answer 46

the min value is 1406 occurs at x=19

Answer 47

\[ \sum_{k=1}^{75} |y - k|\]
the minmum value of this sum occurs at \( y = \frac{75+1}{2} = 38 \) use \(2x = y \)

Answer 48

i see

Answer 49

thanks!!

Answer 50

yw