If the arithmetic mean of the three numbers a, b and c is M and it is given that ab + bc + ca = 0, then what is the arithmetic mean of the three numbers a^2, b^2 and c^2?

Introduction / Context:
This is an algebra-based average question that tests your understanding of how expressions involving a, b and c relate to their mean. You are told that the mean of a, b and c is M and that the sum ab + bc + ca is zero. From this information, you must determine the mean of the squares a^2, b^2 and c^2. The problem requires careful use of algebraic identities.

Given Data / Assumptions:

• The arithmetic mean of a, b, c is M, so (a + b + c) / 3 = M.
• This implies a + b + c = 3M.
• We are also given ab + bc + ca = 0.
• We need the arithmetic mean of a^2, b^2, c^2, which is (a^2 + b^2 + c^2) / 3.

Concept / Approach:
The main algebraic identity used is (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca). Since we know both a + b + c and ab + bc + ca, we can plug them into this identity to find a^2 + b^2 + c^2. Dividing that result by 3 will give the required mean of the squares.

Step-by-Step Solution:
From the given mean, we have (a + b + c) / 3 = M, so a + b + c = 3M.We are given that ab + bc + ca = 0.Use the identity (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca).Substitute a + b + c = 3M and ab + bc + ca = 0 into the identity.Left-hand side: (a + b + c)^2 = (3M)^2 = 9M^2.Right-hand side: a^2 + b^2 + c^2 + 2 * 0 = a^2 + b^2 + c^2.Therefore, a^2 + b^2 + c^2 = 9M^2.The mean of a^2, b^2, c^2 is (a^2 + b^2 + c^2) / 3 = 9M^2 / 3 = 3M^2.

Verification / Alternative check:
To verify, we can test with simple numbers that satisfy ab + bc + ca = 0 and a + b + c = 3M. For example, choose a = t, b = -t, c = 3M so that ab + bc + ca becomes -t^2 + t(3M) - t(3M) = -t^2. To make this zero, choose t = 0, giving a = 0, b = 0, c = 3M. Then the mean of a, b, c is M and ab + bc + ca = 0. The squares are 0, 0, 9M^2, whose mean is 3M^2, matching our formula.

Why Other Options Are Wrong:
Option a, M, ignores the effect of squaring the numbers and the zero condition on ab + bc + ca. Option c, 2(M/3), and option d, M^2, do not arise from the correct algebraic identity. Option e, 3M, treats the sum of the squares as if it scaled linearly with M, which is not correct since we are dealing with squares. Only 3M^2 is consistent with the identity and the given condition.

Common Pitfalls:
Students sometimes forget the correct form of the expansion (a + b + c)^2 or mis-handle the term 2(ab + bc + ca). Others try to guess the answer from dimensional thinking without using the given ab + bc + ca = 0 condition. Always recall the full identity and substitute systematically to avoid errors.

Final Answer:
The arithmetic mean of a^2, b^2 and c^2 is 3M^2.