Problem 17 - Calculus for Biology and Medicine

Question

In Problems 17-24, find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}\). $$ 1,4,9,16,25 $$

Step-by-Step Solution

Answer

The next four terms are 36, 49, 64, and 81.

Step 1: Identify the Sequence

The given sequence is 1, 4, 9, 16, 25. These numbers are perfect squares: 1 is \(1^2\), 4 is \(2^2\), 9 is \(3^2\), 16 is \(4^2\), and 25 is \(5^2\).

Step 2: Determine the Formula

Since the numbers in the sequence correspond to perfect squares, the general term of the sequence \(a_n\) can be described by the formula \(a_n = (n+1)^2\).

Step 3: Find the Next Four Terms

Using the formula \(a_n = (n+1)^2\), find the next four terms in the sequence: - For \(n = 5\), \(a_5 = (5+1)^2 = 6^2 = 36\)- For \(n = 6\), \(a_6 = (6+1)^2 = 7^2 = 49\)- For \(n = 7\), \(a_7 = (7+1)^2 = 8^2 = 64\)- For \(n = 8\), \(a_8 = (8+1)^2 = 9^2 = 81\).

Key Concepts

Perfect Squares

Perfect squares are numbers obtained by squaring an integer. In mathematical terms, a perfect square is expressed as the product of an integer with itself. For instance, the numbers 1, 4, 9, 16, and 25 are perfect squares because you calculate them by multiplying integers with themselves: \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), and \(5^2 = 25\). Recognizing perfect squares is quite helpful as these numbers form a specific and predictable pattern when placed in order.

• In the sequence, each term represents a perfect square: the square of consecutive integers starting from 1.
• This makes it easier to predict the next terms once you identify the pattern of squaring increasing integers.
Understanding this concept of perfect squares is crucial for deciphering sequences or series similar to the one provided in the exercise.

General Term of a Sequence

The concept of the general term is a key element in understanding sequences. It provides a formula that can be used to determine any term in the sequence without listing all preceding terms. For the exercise at hand, the sequence of numbers is framed as perfect squares, so the general term can be expressed as \(a_n = (n+1)^2\).

• This formula allows you to quickly find any term in the sequence by simply substituting the position number \(n\) into the equation.
• The formula takes into account that the series begins with \(n = 0\), which means when you substitute \(n\) with 0, you get \(1^2 = 1\), the first term of the sequence.

Using this general term is advantageous because it saves time and organizes the approach to determining future terms without recalculating each step.

Mathematical Induction

Mathematical induction is a powerful method of mathematical proof typically used to establish that a statement is true for all natural numbers. It consists of two main parts: the base case and the inductive step. Although it is not explicitly required to solve the originally given sequence problem, the principle of mathematical induction can validate the general term: \(a_n = (n+1)^2\).

• Base Case: Verify that the formula works for the initial term(s). For \(n = 0\), \(a_0 = (0+1)^2 = 1\), which aligns with the sequence.
• Inductive Step: Assume the formula is true for \(n = k\), i.e., \(a_k = (k+1)^2\), and prove it for \(n = k+1\). The next term should validate as follows: \(a_{k+1} = ((k+1)+1)^2 = (k+2)^2\).

By successfully showing that the formula holds for these cases, mathematical induction confirms the reliability of the general term for all terms in the sequence. This technique is an essential part of formal mathematics, ensuring sequences and formulas are consistently applied.