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Average

Formulae -
Average(Avg) = Sum of observations/Number of observations
Average of sets of numbers or Scores = Sum of Numbers/Total set
If n=ma+nb, then average of a and b = n/2m

Average of x observation :
if each observation is increased by a, then new average = x+a
if each observation is decreased by a, then new average = x-a
if each observation is multiplied by a, then new average = ax
if each observation is divided by a, then new average = x+a

Average of numbers :
(1+2+3…n)/n = (n+1)/2 (Avg of Consecutive natural numbers upto n)
(2+4+6…n)/n = (n/2)+1 (Avg of consecutive even numbers upto n)
(1+3+5…n)/n = (n+1)/2 (Avg of consecutive odd numbers upto n)
(12+22+32…n2)/n = (n + 1)(2n + 1)/6 (Avg of square of consecutive natural numbers upto n)
(13+23+33…n3)/n = [ n((n+1)/2)^2 ] (Avg of cubes of consecutive natural no. upto n)
Sum of first n natural numbers = n(n + 1)/2
Sum of first n odd numbers = n2
Sum of first n even numbers = n ( n + 1)

For Example- The average weight of 9 mangoes increases by 20 g if one of them
weighing 120 g is replaced by another. The weight of the new mango is?

Explanation : Total weight increased (9 20) gm = 180 gm.
                                           Weight of new mango = (120 + 180) gm
                                                                                    = 300gm.