Question

2 Induction PaDgKIFR2PQRSTUUWXXZIGH 1) Prove ∀n∈N,∃k∈N s.t. 11 n −2 n=9k (2) A natural \# R is said to be a "Racuchen Number when it can be oppessed as 2 2 k +1 for some natural k. Drave ∀k∈N, R k =∏ i=0k−1R i+2

Answer 1

The student's question requires proving mathematical statements related to natural numbers, divisibility, and series, which would likely involve techniques like mathematical induction or the binomial theorem. However, the reference information provided does not pertain directly to the question.

Step-by-step explanation:

The question asks to prove two mathematical statements involving natural numbers and properties related to divisibility and number series. The first part of the problem requires the proof that for every natural number n, there exists a natural number k such that 11n - 2n = 9k. This is a typical exercise in mathematical induction or divisibility. The second part deals with defining and proving a property of a sequence called a "Racuchen Number" which is represented by certain operations applied over natural numbers and series.

Since the information provided in the reference is not directly relevant to the question, we cannot prove the given statements using that information. However, it is worth mentioning the concept of series expansions like the binomial theorem, which could potentially be used in proofs involving algebraic expressions or studying the properties of natural numbers and sequences.