If each of a, b and c is a non-zero number and `(a)/(b)= (b)/(c )`, show that: `(a + b+c) (a-b + c) = a^(2) + b^(2) + c^(2)`

Answer

To solve the problem, we need to show that:

\[ (a + b + c)(a - b + c) = a^2 + b^2 + c^2 \]

given that \(\frac{a}{b} = \frac{b}{c}\).

### Step 1: Cross-Multiply the Given Ratio

From the ratio \(\frac{a}{b} = \frac{b}{c}\), we can cross-multiply:

\[ a \cdot c = b^2 \]

Let’s label this as Equation (1).

### Step 2: Expand the Left-Hand Side (LHS)

Now we will expand the left-hand side of the equation we want to prove:

\[ (a + b + c)(a - b + c) \]

Using the distributive property (also known as the FOIL method for binomials):

\[ = a(a - b + c) + b(a - b + c) + c(a - b + c) \]

Calculating each term:

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

Combining these results:

\[ = a^2 - ab + ac + ab - b^2 + bc + ac - bc + c^2 \]

Notice that \(-ab\) and \(+ab\) cancel out, and \(+bc\) and \(-bc\) also cancel out:

\[ = a^2 - b^2 + 2ac + c^2 \]

### Step 3: Substitute Using Equation (1)

From Equation (1), we have \(ac = b^2\). Substitute \(ac\) in the expression:

\[ = a^2 - b^2 + 2b^2 + c^2 \]

This simplifies to:

\[ = a^2 + b^2 + c^2 \]

### Conclusion

Thus, we have shown that:

\[ (a + b + c)(a - b + c) = a^2 + b^2 + c^2 \]