We are given a mathematical problem involving the expression (3 + √2)^n. This expression can be written in the form a_n + b_n√2, where a_n and b_n are sequences of rational numbers. Our task is to prove that the limit of a_n divided by b_n√2 equals 1 as n approaches infinity. This is a fascinating problem that combines binomial expansion, conjugate expressions, and asymptotic analysis.
To understand the structure of a_n and b_n, we apply the binomial theorem to expand (3 + √2)^n. The binomial expansion gives us a sum where each term has the form C(n,k) times 3^(n-k) times (√2)^k. The key insight is that terms with even powers of √2 become rational numbers, contributing to a_n, while terms with odd powers of √2 remain irrational, contributing to b_n√2. Let's see this with examples: for n=1, we get 3 + √2. For n=2, we get 11 + 6√2. For n=3, we get 33 + 23√2. This pattern clearly separates into rational and irrational parts.
Now we introduce the conjugate expression (3 - √2)^n. When we expand this using the binomial theorem, we get the same rational coefficients as before, but the irrational parts have opposite signs. So (3 - √2)^n equals a_n minus b_n√2. The key insight is that 3 - √2 is approximately 1.586, which is less than 2. This means that as n approaches infinity, (3 - √2)^n approaches zero. This property of the conjugate expression will be crucial for our limit analysis.
Now we establish the fundamental relationships between the original and conjugate expressions. When we add (3 + √2)^n and (3 - √2)^n, the irrational parts cancel out, giving us 2a_n. When we subtract them, the rational parts cancel out, giving us 2b_n√2. This leads to explicit formulas: a_n equals the sum divided by 2, and b_n equals the difference divided by 2√2. We can verify this with our n=2 example: a_2 equals 11, and b_2 equals 6, which matches our earlier calculations.
Now we complete the rigorous proof. We can write the ratio a_n over b_n√2 as a fraction involving the conjugate terms. The key insight is that the ratio (3-√2)/(3+√2) equals (3-√2)² divided by 7, which is less than 1. Therefore, this ratio raised to the power n approaches 0 as n approaches infinity. This means our limit becomes (1+0)/(1-0) = 1. The fundamental insight is that the conjugate expression's exponential decay is what drives this limiting behavior, making the ratio approach exactly 1.