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When a sequence is called strictly increasing, each term in the sequence is greater than the one before it. Think of it like climbing a staircase where every step you take is higher than the last. Mathematically, if you have a sequence \( a_n \), it is strictly increasing if \( a_{n+1} > a_n \) for all \( n \).

To determine if a sequence is strictly increasing, you check whether the related function’s derivative is positive over the interval you are interested in. If the derivative \( f'(x) > 0 \) for all \( x \) in the interval, it means the function, and therefore the sequence at integer points, climbs upwards continuously.

• Checking the sign of the derivative can help you decide if you're going up or down.
• Positive derivative means increasing sequence.

Understanding this concept ensures you're looking at the changes between terms correctly and can accurately determine their behavior.

A sequence is strictly decreasing when every term is less than the one before it. It's like descending a staircase where each step down brings you lower. In mathematical terms, a sequence \( a_n \) is strictly decreasing if \( a_{n+1} < a_n \) for all \( n \).

The method to prove a strictly decreasing sequence involves checking the sign of the derivative of the associated function. When the derivative \( f'(x) < 0 \), the function descends, and so does the sequence.

• Negative derivative indicates the sequence is strictly decreasing.
• This method simplifies the problem by translating it into the language of calculus through differentiation.

This allows a clear determination of the sequence's behavior.

The quotient rule is a key concept in calculus used to differentiate functions that are ratios of two differentiable functions. The rule is expressed as: if \( h(x) = \frac{u(x)}{v(x)} \), then \( h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).

This might seem complex at first glance, but breaking it down:

• Identify \( u(x) \) and \( v(x) \), the functions in the numerator and denominator.
• Find their derivatives, \( u'(x) \) and \( v'(x) \).
• Plug these into the quotient rule formula to find \( h'(x) \).

This systematic approach provides the derivative of the quotient, allowing further analysis such as determining if a function is increasing or decreasing. The quotient rule not only helps with differentiation but also simplifies evaluating specific sequences related to the given functions.