Problema Solution
Group A has 10 numbers with an average of 100. Group B has 20 numbers with an average of 400. Group C has 30 numbers with an average of 900. This pattern goes on with all other letter groups (D, E, F, G, and so on) with that pattern, ending with Group Z having 260 numbers with an average of 260^2. All of the groups are combined to make Group Sigma. Find the average of the numbers in group Sigma.
Answer provided by our tutors
Group A: the sum of the 10 numbers with an average of 100 is: 10*100 = 10^3
Group B: the sum of the 20 numbers with an average of 400 is: 20*400 = 20^3
Group C: the sum of the 30 numbers with an average of 900 is: 30*900 = 30^3
....
Group Z: the sum of the 260 numbers with an average of 260^2 is: 260*260^2 = 260^3
The average of group Sigma is:
(10^3 + 20^3 + 30^3 +.... + 260^3)/(10 + 20 + 30 + ...260) =
= 10^3(1^3 + 3^3 + 3^3 + ... + 26^3)/(10(1 + 2 + 3 + ...+26)) =
= 10^2(1^3 + 3^3 + 3^3 + ... + 26^3)/(1 + 2 + 3 + ...+26)
using the formulas:
(1^3 + 2^3 + ... + n^3) = [n^2 (n + 1)^2]/4
(1 + 2 + ... + n) = [n(n+1)]/2
and in our case n = 26 we have:
(1^3 + 2^3 + ... + 26^3) = [26^2 (26 + 1)^2]/4 = (13^2) (27^2)
(1 + 2 + ... + 26) = [26(26+1)]/2 = 13*27
10^2(1^3 + 3^3 + 3^3 + ... + 26^3)/(1 + 2 + 3 + ...+26) =
= (10^2) (13^2) (27^2)/(13*27) =
= 100*13*27 =
= 35100
he average of the numbers in group Sigma is 35100.