Understanding the Arithmetic Mean

The arithmetic mean (or average) of a set of numbers is found by summing the numbers and dividing by the count of numbers in the set.

Analyzing the Given Information

We are given two pieces of information:

1. The arithmetic mean of a, b, and c is \(\rm \frac{M}{3}\).
2. The relationship between the reciprocals is \(\rm \frac{1}{a} + \frac{1}{b} = -\frac{1}{c}\).

Step-by-Step Solution

Deriving Relationships from the Given Mean

From the definition of arithmetic mean:

\(\rm \frac{a + b + c}{3} = \frac{M}{3}\)

Multiplying both sides by 3 gives:

\(\rm a + b + c = M\)

Simplifying the Reciprocal Relationship

Start with the second given condition:

\(\rm \frac{1}{a} + \frac{1}{b} = -\frac{1}{c}\)

Combine the terms on the left side using a common denominator (ab):

\(\rm \frac{b + a}{ab} = -\frac{1}{c}\)

Cross-multiply:

\(\rm c(a + b) = -ab\)

Expand the left side:

\(\rm ac + bc = -ab\)

Rearrange the terms to one side:

\(\rm ab + bc + ac = 0\)

Finding the Mean of Squares

We need to find the arithmetic mean of a2, b2, and c2, which is \(\rm \frac{a^2 + b^2 + c^2}{3}\).

We know the identity:

\(\rm (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ac)\)

Substitute the known values into this identity:

• a + b + c = M
• ab + bc + ac = 0

So, the equation becomes:

\(\rm (M)^2 = a^2 + b^2 + c^2 + 2(0)\)

\(\rm M^2 = a^2 + b^2 + c^2\)

Calculating the Final Arithmetic Mean

Now substitute this result back into the formula for the arithmetic mean of the squares:

Arithmetic Mean = \(\rm \frac{a^2 + b^2 + c^2}{3}\)

Arithmetic Mean = \(\rm \frac{M^2}{3}\)

Conclusion

The arithmetic mean of a2, b2, and c2 is \(\rm \frac{M^2}{3}\).