Zeek The Greek Homework Set 10 Assigned March 21 Due Wednesd

Zeek The Greekhomework Set 10assigned Monday March 21due Wednesda

Define the following sequence by recursion: a0 = 1 and for all integers n > 0, an = 1 + Σ_{i=0}^{n−1} 2a_i. Show by induction that for all integers n ≥ 0, an = 3^n. Use the geometric series formula proven in class: Σ_{k=m}^{n} r^k = (r^m − r^{n+1}) / (1− r).

Define the sequence by recursion: a0 = 2, and for all integers n > 0, an = 2 + 2 Σ_{i=0}^{n−1} a_i. Show by induction that for all integers n ≥ 0, an ≤ 4n+1. Determining an exact formula is possible but not necessary.

Define two sequences (an) and (bn) by recursion: a0 = 1, and an = √2 · an−1 for n > 0; b0 = 5, b1 = 5, and bn = bn−1 + bn−2 for n > 1. Show that for all n ≥ 0, bn > an.

Show by strong induction that for all positive integers n, there exist integers a and b such that n = 3ab and 3 divides b.

Show that the representation from (4) is unique: if n = 3ab = 3cd, with integers a, b, c, d, and 3 divides (b− d), then a = c and b = d.

Show that for all integers n ≥ 43, there exist nonnegative integers a and b such that n = 6a + 7b.

Show that for all integers n ≥ 0, if n is divisible by 4, then 5 divides 2n+2 + 3n+4.

Paper For Above instruction

This paper explores several properties of integer sequences and their relationships, employing various proof techniques including induction, geometric series analysis, and combinatorial reasoning. The problems presented involve defining sequences recursively, establishing bounds and identities, and proving uniqueness and divisibility properties without relying on prime factorization. Through these exercises, the goal is to deepen understanding of recursive definitions, inequalities, sequence comparisons, and algebraic uniqueness, all cornerstones of discrete mathematics and number theory.

The first problem involves a recursively defined sequence where each term is expressed in terms of the sum of previous terms multiplied by 2, plus an initial value of 1. Using induction, it’s demonstrated that the explicit formula for this sequence is an exponential function, specifically an exponential growth at rate 3. The proof leverages the geometric series sum to simplify the recursive sum, leading to the conclusion that an = 3^n for all n ≥ 0.

In the second problem, the sequence defined similarly by recursion with an initial value of 2 is shown—via induction—that it is bounded above by a linear function, 4n + 1. Recognizing the recursive pattern and employing the induction hypothesis allows an approximate bound without explicitly solving for the exact closed-form expression. This inequality captures the growth rate of the sequence and provides an understanding of its asymptotic behavior.

The third problem combines two sequences: one defined by multiplicative recursion involving √2, and the other by a standard Fibonacci-like recurrence relation. The task is to prove that the sequence with Fibonacci recursion dominates the geometric sequence at all steps. This involves comparing terms recursively and establishing the inequality bn > an for all n, possibly using induction or direct comparison, which showcases the relationship between exponential and Fibonacci-like sequences.

The fourth problem asks for a proof—via strong induction—that any positive integer n can be expressed as n = 3ab, with 3 dividing b. This is inspired by the structure of prime factorization, but without directly invoking it. The proof involves constructing appropriate divisors and leveraging the divisibility properties to decompose n into the product form that satisfies the divisibility condition, thereby illustrating the underlying number-theoretic structure.

The fifth problem proves the uniqueness of the representation n = 3ab, ensuring that if two such representations yield the same n and satisfy the divisibility condition, then their components must coincide. The proof demonstrates that the factorization form is canonical and that any alternative decomposition must be identical, emphasizing the importance of unique factorization concepts in a restricted context.

The sixth problem involves demonstrating that every integer n ≥ 43 can be expressed as a linear combination of 6 and 7 with nonnegative coefficients. The proof uses induction or the Frobenius coin problem principle, establishing base cases and then inductively constructing larger numbers from known combinations, thus verifying the coverage of integers beyond a certain threshold.

Lastly, the seventh problem examines divisibility properties involving n divisible by 4. Specifically, it shows that under this condition, the sum 2n+2 + 3n+4 is divisible by 5. The proof involves algebraic manipulations and modular arithmetic, illustrating how divisibility rules and properties of modular congruences apply to algebraic expressions involving multiples and sums.

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