(a) Determine whether the sequence is increasing, decreasing, or not monotonic. (Assume that $n$ begins with $1$.) $a_n = \frac{3n - 4}{4n + 3}$

• increasing
• decreasing
• not monotonic

(b) Is the sequence bounded?

• bounded
• not bounded

Neetesh Kumar | December 31, 2024

Calculus Homework Help

This is the solution to Math 1c
Assignment: 11.1 Question Number 37
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Step-by-step solution:

Part (a): Determine monotonicity:

Step 1: Compute $a_{n+1}$ and compare it to $a_n$:

The sequence is $a_n = \frac{3n - 4}{4n + 3}$.

To determine if it is increasing or decreasing, we analyze the difference $a_{n+1} - a_n$.

Step 2: Write $a_{n+1}$:

Substitute $n+1$ into the formula for $a_n$:

$a_{n+1} = \frac{3(n+1) - 4}{4(n+1) + 3} = \frac{3n + 3 - 4}{4n + 4 + 3} = \frac{3n - 1}{4n + 7}$

Step 3: Analyze $a_{n+1} - a_n$:

Compute the difference:

$a_{n+1} - a_n = \frac{3n - 1}{4n + 7} - \frac{3n - 4}{4n + 3}$

Combine under a common denominator:

$a_{n+1} - a_n = \frac{(3n - 1)(4n + 3) - (3n - 4)(4n + 7)}{(4n + 7)(4n + 3)}$

Expand the numerator:

$(3n - 1)(4n + 3) = 12n^2 + 9n - 4n - 3 = 12n^2 + 5n - 3$

$(3n - 4)(4n + 7) = 12n^2 + 21n - 16n - 28 = 12n^2 + 5n - 28$

Subtract the two terms:

$a_{n+1} - a_n = \frac{(12n^2 + 5n - 3) - (12n^2 + 5n - 28)}{(4n + 7)(4n + 3)}$

Simplify the numerator:

$12n^2 + 5n - 3 - 12n^2 - 5n + 28 = 25$

Thus:

$a_{n+1} - a_n = \frac{25}{(4n + 7)(4n + 3)}$

Step 4: Sign of $a_{n+1} - a_n$:

The denominator $(4n + 7)(4n + 3) > 0$ for all $n \geq 1$, and the numerator $25 > 0$.

Therefore:

$a_{n+1} - a_n > 0$

This shows that the sequence is increasing.

Part (b): Determine boundedness:

Step 1: Check for upper and lower bounds:

The sequence is:

$a_n = \frac{3n - 4}{4n + 3}$

For large $n$, divide numerator and denominator by $n$:

$a_n = \frac{3 - \frac{4}{n}}{4 + \frac{3}{n}} \to \frac{3}{4} \quad \text{as } n \to \infty$

For $n = 1$, $a_1 = \frac{3(1) - 4}{4(1) + 3} = \frac{-1}{7} = -\frac{1}{7}$. Therefore, the sequence is bounded below by $-\frac{1}{7}$ and above by $\frac{3}{4}$.

Step 2: Conclusion:

The sequence is bounded.

Final Answers:

(a) The sequence is $\boxed{\text{increasing}}$

(b) The sequence is $\boxed{\text{bounded}}$

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