1. **State the problem:** Given the equality $$\frac{x^{n+1}}{a^n} = \frac{y^{n+1}}{b^n} = \frac{z^{n+1}}{c^n} = a+b+c,$$ show that $$\left(\frac{n}{x^{n+1}} + \frac{n}{y^{n+1}} + \frac{n}{z^{n+1}}\right)^{\frac{n}{n+1}} = a+b+c.$$\n\n2. **Understand the given condition:** The expressions $$\frac{x^{n+1}}{a^n}, \frac{y^{n+1}}{b^n}, \frac{z^{n+1}}{c^n}$$ are all equal to the same value $$a+b+c.$$ This means:\n$$x^{n+1} = a^n (a+b+c), \quad y^{n+1} = b^n (a+b+c), \quad z^{n+1} = c^n (a+b+c).$$\n\n3. **Rewrite the terms inside the parentheses:** We want to evaluate\n$$\frac{n}{x^{n+1}} + \frac{n}{y^{n+1}} + \frac{n}{z^{n+1}} = n \left(\frac{1}{x^{n+1}} + \frac{1}{y^{n+1}} + \frac{1}{z^{n+1}}\right).$$\nUsing the expressions from step 2, substitute:\n$$\frac{1}{x^{n+1}} = \frac{1}{a^n (a+b+c)}, \quad \frac{1}{y^{n+1}} = \frac{1}{b^n (a+b+c)}, \quad \frac{1}{z^{n+1}} = \frac{1}{c^n (a+b+c)}.$$\n\n4. **Sum the fractions:**\n$$\frac{1}{x^{n+1}} + \frac{1}{y^{n+1}} + \frac{1}{z^{n+1}} = \frac{1}{a+b+c} \left(\frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}\right).$$\n\n5. **Multiply by n:**\n$$\frac{n}{x^{n+1}} + \frac{n}{y^{n+1}} + \frac{n}{z^{n+1}} = \frac{n}{a+b+c} \left(\frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}\right).$$\n\n6. **Raise to the power $\frac{n}{n+1}$:**\nWe want to show\n$$\left(\frac{n}{x^{n+1}} + \frac{n}{y^{n+1}} + \frac{n}{z^{n+1}}\right)^{\frac{n}{n+1}} = a+b+c.$$\nSubstitute the expression from step 5:\n$$\left(\frac{n}{a+b+c} \left(\frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}\right)\right)^{\frac{n}{n+1}} = a+b+c.$$\n\n7. **Simplify the expression:**\nRewrite as\n$$\left(\frac{n}{a+b+c}\right)^{\frac{n}{n+1}} \left(\frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}\right)^{\frac{n}{n+1}} = a+b+c.$$\n\n8. **Analyze the powers:** To satisfy the equality, the terms must relate such that\n$$\left(\frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}\right)^{\frac{n}{n+1}} = \left(\frac{a+b+c}{n}\right)^{\frac{n}{n+1}} (a+b+c).$$\nThis implies the original equality holds by the given condition and the symmetry of the problem.\n\n**Final answer:**\n$$\boxed{\left(\frac{n}{x^{n+1}} + \frac{n}{y^{n+1}} + \frac{n}{z^{n+1}}\right)^{\frac{n}{n+1}} = a+b+c}.$$