(an)=((a^(n)+b^(n)+c^(n)/3)^( 1/n ) convergence of the sequence

To determine the convergence of the sequence \( a_n = \left( \frac{a^n + b^n + c^n}{3} \right)^{\frac{1}{n}} \), we analyze the behavior of the sequence as \( n \to \infty \).

### Case 1: \( a = b = c \)

If \( a = b = c \), then:
\[
a_n = \left( \frac{a^n + a^n + a^n}{3} \right)^{\frac{1}{n}} = \left( \frac{3a^n}{3} \right)^{\frac{1}{n}} = a.
\]
In this case, the sequence is constant and converges to \( a \).

### Case 2: \( a, b, c \) are distinct

Assume that \( a, b, c \) are distinct. Without loss of generality, let \( a > b > c \). Then, as \( n \to \infty \), the term \( a^n \) will dominate over \( b^n \) and \( c^n \), because the exponential function grows faster for larger bases. More precisely:
\[
\lim_{n \to \infty} \frac{a^n + b^n + c^n}{3} = \lim_{n \to \infty} \frac{a^n (1 + (\frac{b}{a})^n + (\frac{c}{a})^n)}{3}.
\]
Since \( \frac{b}{a} < 1 \) and \( \frac{c}{a} < 1 \), the terms \( (\frac{b}{a})^n \) and \( (\frac{c}{a})^n \) tend to zero as \( n \to \infty \). Hence, we have:
\[
\lim_{n \to \infty} \frac{a^n + b^n + c^n}{3} = \frac{a^n}{3} \left( 1 + 0 + 0 \right) = \frac{a^n}{3}.
\]
Thus,
\[
a_n \sim \left( \frac{a^n}{3} \right)^{\frac{1}{n}} = \frac{a}{3^{1/n}}.
\]
As \( n \to \infty \), \( 3^{1/n} \to 1 \), so:
\[
\lim_{n \to \infty} a_n = a.
\]

### Conclusion

In all cases, the sequence \( a_n \) converges to \( \max(a, b, c) \). Therefore, the sequence converges to the largest of \( a \), \( b \), or \( c \).

Would you like a detailed explanation on any part of this, or do you have further questions?

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Here are five related questions:
1. How would the sequence behave if one of the variables \( a \), \( b \), or \( c \) is zero?
2. How does the convergence rate of the sequence depend on the values of \( a \), \( b \), and \( c \)?
3. What would happen if the terms were divided by a constant other than 3?
4. Can the method of dominant terms be applied to more than three terms?
5. What is the behavior of the sequence if \( a \), \( b \), and \( c \) are negative?

**Tip**: In sequences, focusing on dominant terms is often key when determining convergence, especially in cases involving powers and exponential growth.

The query submitted by the user: an = ((a^n + b^n + c^n) / 3)^(1/n) convergence of the sequence

Analyze the convergence of the sequence an = ((a^n + b^n + c^n) / 3)^(1/n) as n approaches infinity. Learn how dominant terms in exponential growth lead to the sequence converging to the largest value among a, b, and c. Suitable for advanced high school and undergraduate students.

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