Problem 15 - Thomas Calculus

Question

Find a formula for the \(n\)th term of the sequence. $$ 1,-4,9,-16,25, \dots $$

Step-by-Step Solution

Answer

The nth term formula is \(a_n = (-1)^{n+1} \times n^2\).

Step 1: Identify the Pattern

Look at how the sequence is formed: \(1, -4, 9, -16, 25, \dots\). Notice that the absolute values form a pattern of square numbers: \(1^2, 2^2, 3^2, 4^2, 5^2, \dots\). However, there is an alternating sign pattern: positive, negative, positive, negative, \dots.

Step 2: Determine the Base Term Pattern

Recognize that the absolute values of the terms correspond to perfect squares: \(1, 4, 9, 16, 25, \dots\), which are \(1^2, 2^2, 3^2, 4^2, 5^2, \dots\). This means the nth term without considering signs is \(n^2\).

Step 3: Account for the Sign Alternation

To include the alternating signs, notice that the sequence goes positive, negative, positive, negative, etc. An expression that alternates signs like \((-1)^{n+1}\) can be used. Multiply this by the base term pattern: \((-1)^{n+1} \times n^2\).

Step 4: Formulate the General Term

Combine the derived elements. The formula for the nth term of the sequence is \(a_n = (-1)^{n+1} \times n^2\). This accounts for both the alternating signs and the sequence of squares.

Key Concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. In simpler terms, a perfect square is obtained when an integer is multiplied by itself. For example, the number 9 is a perfect square because it can be written as \(3^2\), which means \(3 \times 3\). Similarly, 16 is a perfect square because it's \(4^2\), which is \(4 \times 4\).

In our sequence, the absolute values are perfect squares: \(1, 4, 9, 16, 25, \dots\). This shows that they are represented by \(1^2, 2^2, 3^2, 4^2, 5^2, \dots\). Understanding that the sequence forms perfect squares is key to identifying the underlying pattern of the sequence.

Alternating Series

An alternating series is a sequence in which the sign of each term alternates between positive and negative. In our sequence, the signs go positive, negative, positive, negative, and so on. This is recognized as a sign pattern, which cycles between these two values.

The pattern in the original sequence is \(1, -4, 9, -16, 25, \dots\). To model this alternating sign, we use the expression \((-1)^{n+1}\). This expression works because:

• When \(n\) is odd, \((-1)^{n+1}\) is \(1\).
• When \(n\) is even, \((-1)^{n+1}\) is \(-1\).

This trick helps in constructing formulas for sequences with alternating signs.

Nth Term Formula

The nth term formula is a way to find any term in a sequence without listing all the previous terms. It gives us a mathematical expression to determine the term based on its position in the sequence.

In this exercise, we derived the nth term for the given sequence as \(a_n = (-1)^{n+1} \times n^2\). Breakdown of the formula includes:

• \((-1)^{n+1}\) provides the alternating sign.
• \(n^2\) gives the sequence's perfect square pattern.

Combine these to tackle exercises where both numeric patterns and alternating signs need encoding into a single expression.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers, defined by specific rules or patterns. Understanding sequences involves looking at patterns in the number's arrangement, such as finding a common difference in arithmetic sequences or a common ratio in geometric sequences.

Our sequence showcases both perfect squares and alternating signs. This demonstrates how different sequence rules can combine. Sequences are foundational in mathematics because they represent patterns and structures used extensively in higher math concepts.

When you identify patterns within a sequence - like alternating signs and perfect squares - you can formulate a general rule for the sequence. This knowledge aids in solving related mathematical problems and understanding the nature of number patterns.