The mean of 10 observation is x. If the first item is increased by 1, second by 2, and so on then what is the new mean?

Question

The length of a rectangle exceeds its width by 5m. If the width is increased by 1m and the length is decreased by 2m, the area of the new rectangle is 4 sq.m less than the area of the original rectangle. Find the dimensions of the original rectangle.

Answer 1

If the average is x. sum of all the ten numbers ought to be 10x. If one is added to first number, 2 to the next and so on we add 1+2+3+4+5+6+7+8+9+10 to the total which becomes 10x+55 whose average is x+5.5. In case instead of 10 numbers, we have say 50 numbers which are not easy to add manually use the sum of the series formula of Sn=(n/2)*(2*a+(n-1)*d)

Comments

What makes you think the numbers obey arithmetic progression???
Why do you think the difference btn any two consecutive numbers is constant???

Answer 2

Let the numbers be a, b, c, d, e, f, g, h, i and j
Then,
(a+b+c+d+e+f+g+h+i+j)/10 =x.................................(1)

If a is increased by 1, b by 2 and so on, then

New mean = {(a+1) + (b+2) + (c+3) + (d+4) + (e+5) + (f+6) + (g+7) + (h+8) + (i+9) + (j+10)}/10

                = **{(a+b+c+d+e+f+g+h+i+j) + (1+2+3+4+5+6+7+8+9+10)}/10**

               = **(a+b+c+d+e+f+g+h+i+j)/10 + 55/10**...........substitute eqn(1) to this eqn we have,

               = **x + 5.5**

So the new mean (n.m) is,

**

n.m = x + 5.5

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Comments

I have never assumed that the difference between any  two consecutive numbers is constant but that the difference between the additions of 1,2,....10 is constant and 1.