Question 83
Suppose that $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3, \dots$ are two sequences of real numbers. These sequences are said to be Co-Algebraic if a non-zero two-variable polynomial $P(x, y)$ with real coefficients exists such that for each natural number $n$, $P(a_n, b_n) = 0$.
a) Prove that sequences $n$ and $2^n$ (for each natural number $n$) are not Co-Algebraic.
b) Are sequences $2^n$ and $3^n$ (for each natural number $n$) Co-Algebraic?
c) Suppose that $f(x, y)$ is a non-zero two-variable polynomial with real coefficients. Prove that there exists a natural number $n$ such that $f(2^n, 3^n)$ is not divisible by $5^n$.
a) Prove that sequences $n$ and $2^n$ (for each natural number $n$) are not Co-Algebraic.
b) Are sequences $2^n$ and $3^n$ (for each natural number $n$) Co-Algebraic?
c) Suppose that $f(x, y)$ is a non-zero two-variable polynomial with real coefficients. Prove that there exists a natural number $n$ such that $f(2^n, 3^n)$ is not divisible by $5^n$.