Problem 5 - Precalculus with Limits

Question

In Exercises 5 - 10, find \( P_{k + 1} \) for the given \( P_k \). \( P_k = \dfrac{5}{k\left(k + 1\right)} \)

Step-by-Step Solution

Answer

\( P_{k + 1} = \dfrac{5}{\left(k + 1\right)\left(k + 2\right)} \)

Step 1: Substitution

Given that \( P_k = \dfrac{5}{k\left(k + 1\right)} \), substitute \( k + 1 \) for \( k \) in the formula to calculate \( P_{k + 1} \). Thus, \( P_{k + 1} = \dfrac{5}{\left(k + 1\right)\left(\left(k + 1\right) + 1\right)} \).

Step 2: Simplify

Simplify the expression \( P_{k + 1} = \dfrac{5}{\left(k + 1\right)\left(\left(k + 1\right) + 1\right)} \) to its simplest form. Therefore, \( P_{k + 1} = \dfrac{5}{\left(k + 1\right)\left(k + 2\right)} \).

Key Concepts

Mathematical Induction

Mathematical induction is a powerful tool often employed in mathematics to prove sequence or series-related propositions. It involves two main steps: the base case and the inductive step. In the base case, you verify that a statement holds for an initial value, often the first term in a sequence, typically denoted as when 'k' equals one or zero. Following this, the inductive step requires you to assume the statement holds for 'k' and then prove it must also hold for 'k + 1'.

For instance, when given a sequence formula like the problem at hand, mathematical induction can be used to show that the formula is valid for all natural numbers 'k'. However, our goal here isn't to prove the formula works for every 'k' but to find the next term, which is a necessary skill in understanding how sequences progress and are constructed.

Sequence Formulas

Sequence formulas are mathematical expressions that define a pattern for generating the elements of a sequence. In this exercise, the sequence is defined by the formula given for 'P_k'. Understanding the notation and how to manipulate these formulas is crucial for anyone studying sequences and series. The formula given can be thought of as a machine that takes a value of 'k' and produces the corresponding term in the sequence. This is why replacing 'k' with 'k + 1' allows us to find the next term, 'P_{k + 1}', illustrating the sequence’s nature of progressing from one term to another.

The sequence formula often reflects some kind of patterning or relationship between terms, as is seen with the term involving 'k(k + 1)', which suggests a pattern of consecutive integer multiplication. Knowing how to work with these expressions is essential for effectively analyzing the behavior of sequences over time.

Simplifying Expressions

Simplifying expressions is a vital skill in mathematics, reducing complexity and making it easier to work with equations. In the context of sequences, simplifying the formula for 'P_{k + 1}' means rewriting it in the most condensed form possible without changing its value. In the provided example, the simplification process involves expanding the terms in the denominator and then combining like terms, if any, which was unnecessary in this case as the expression was already given in factored form.

For sequences, simplified formulas allow a clearer view of the relationship between terms and can sometimes reveal patterns that are not immediately obvious in the original formulation. This is crucial for predicting the behavior of the sequence and for computational efficiency when working with longer sequences.