Solving Mathematical Problems with Logical Reasoning and Algebraic Techniques

In this article, we delve into detailed solutions for complex mathematical problems using logical reasoning and algebraic techniques. We'll cover three distinct problems, each requiring a unique approach to find a solution. Let's explore these problems step by step.

1. Moose Population Model

Given that the number of moose M on the island of Newfoundland is modeled by the formula ( M 75,000 times 5^{-t} ), where ( t ) is time in years, we need to determine how many years it will take for the moose population to grow to 100,000 animals.

Solution: Since the population is increasing, we can use the formula for exponential growth:

[ text{Increased population} frac{text{Original population} times (1 text{Rate of increase})^t}{10^{5-10^{-t}}}75,000 times 5^{-t} ]

To isolate ( t ), we take the logarithm of both sides:

[ 5t log 2 log 4 log 1000 ]

Solving for ( t ), we get:

[ t approx frac{log 4 log 1000}{5 log 2} approx frac{0.602 3}{0.699} approx 5.153 ]

This means it takes approximately 5 years, 1 month, and 25 days for the moose population to reach 100,000 animals.

2. System of Linear Equations

The equations given are:

[ 35^56 151872 ]

[ 55^6 253094 ]

[ 56^7 303585 ]

[ 55^3 251573 ]

To solve for these, we need to find a consistent set of equations with three unknowns. Here, we see that:

[ 3 - 55^56 - 55^56 101222 Rightarrow 155^56 50611 ]

[ 55^56 - 55^56 55^56 1521 Rightarrow 55^56 507 ]

[ 56^56 - 55^56 55^56 50491 Rightarrow 55^56 49984 ]

Thus, we can use these to form the equation:

[ 95^56 455^56 755^56 50611 times 9 49984 times 4 7 times 507 658984 ]

This confirms that the values are consistent.

3. Summation and Mathematical Optimization

We are given the sum:

[ sum_{k1}^n a_k frac{n(n 1)}{2} ]

and need to simplify the expression:

[ sum_{k1}^n frac{k^2 - 1}{a_k^2 a_k 1} ]

First, we assume ( a_k k ). Then the expression simplifies to:

[ sum_{k1}^n frac{k^2 - 1}{k^2 k 1} sum_{k1}^n k frac{n(n 1)}{2} ]

This confirms that the option B (( n(n 1)/2 )) is the right answer. Additionally, we can prove that ( a_k - a_{k-1} k ) from the given expression.

Conclusion: Understanding the problem and applying logical reasoning and algebraic techniques are crucial for solving complex mathematical problems. These techniques help us simplify expressions, find consistent solutions, and optimize results.