Problem 2 - Calculus for Biology and Medicine

Question

Determine the values of the sequence \(\left\\{a_{n}\right\\}\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=3 n^{2} $$

Step-by-Step Solution

Answer

The sequence values are 0, 3, 12, 27, 48, and 75 for \(n = 0, 1, 2, 3, 4, 5\).

Step 1: Understand the sequence formula

The sequence is defined by the formula \( a_n = 3n^2 \). This formula means that each term in the sequence is calculated by squaring the term index \( n \), then multiplying it by 3.

Step 2: Calculate the first term \(a_0\)

Substitute \( n = 0 \) into the formula to find the first term: \( a_0 = 3(0)^2 = 3 imes 0 = 0 \).

Step 3: Calculate the second term \(a_1\)

Substitute \( n = 1 \) into the formula to find the second term: \( a_1 = 3(1)^2 = 3 imes 1 = 3 \).

Step 4: Calculate the third term \(a_2\)

Substitute \( n = 2 \) into the formula to find the third term: \( a_2 = 3(2)^2 = 3 imes 4 = 12 \).

Step 5: Calculate the fourth term \(a_3\)

Substitute \( n = 3 \) into the formula to find the fourth term: \( a_3 = 3(3)^2 = 3 imes 9 = 27 \).

Step 6: Calculate the fifth term \(a_4\)

Substitute \( n = 4 \) into the formula to find the fifth term: \( a_4 = 3(4)^2 = 3 imes 16 = 48 \).

Step 7: Calculate the sixth term \(a_5\)

Substitute \( n = 5 \) into the formula to find the sixth term: \( a_5 = 3(5)^2 = 3 imes 25 = 75 \).

Key Concepts

Understanding the Sequence Formula

In mathematics, an arithmetic sequence is a sequence of numbers where each term after the first is found by adding a constant to the previous term. However, the formula for this sequence: \( a_n = 3n^2 \), where \( n \) is the index, is not an arithmetic sequence but a quadratic one. Here, the formula tells us that each term of the sequence is determined by taking the index \( n \), squaring it, and then multiplying by 3.

To decipher the sequence, follow these pointers:

• The index \( n \) represents the position of the term in the sequence.
• The operation \( n^2 \) means you multiply \( n \) by itself.
• The resultant value after squaring is then multiplied by 3.

This pattern helps to calculate each term of a sequence systematically. For instance, if \( n = 2 \), then \( a_2 = 3 \times (2^2) = 3 \times 4 = 12 \). Hence, using the sequence formula, the consistent pattern can be revealed, yielding each term accordingly.

Exploring Squared Terms

Squared terms are fundamental in mathematics. When you square a number, you multiply that number by itself. In our exercise, squaring is a central operation. For instance, given the sequence formula \( a_n = 3n^2 \), squaring \( n \) gives us \( n^2 \). This means if \( n = 2 \), squaring \( n \) results in \( 4 \), because \( 2 \times 2 = 4 \).

Here’s why squared terms are important:

• Squaring constraints the sequence to grow quadratically, not linearly.
• Each increase in \( n \) results in the difference between consecutive terms growing larger, not by a constant value.
• If you were to plot this sequence, it would form a parabola, illustrating its quadratic nature.

Thus, the use of squared terms not only affects the calculation of individual terms but also dictates the overall behavior and characteristics of the sequence. Being familiar with this can improve how you approach and understand different mathematical problems.

The Role of Index Substitution

Index substitution is a straightforward, yet crucial step in sequence calculations and helps in deriving specific terms. In this sequence, the symbol \( n \) acts as the index. To find the term value at a specific index, substitute \( n \) with the corresponding number, like \( n = 0, 1, 2, \ldots \). This substitution is central to determining each unique term in the sequence.

What to remember about index substitution:

• Start with the initial index from 0 and proceed forward to whatever term is needed.
• Replace \( n \) in the sequence formula with the desired index value to calculate terms, one at a time.
• Ensure the substitution happens for each term you want to compute, maintaining accuracy in calculations.

By following index substitution meticulously, you can compute the sequence efficiently and avoid errors. For example, with \( n=3 \), substituting gives \( a_3 = 3 \times (3^2) = 3 \times 9 = 27 \). Each step brings clarity and structure to solving sequence problems.