Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
1, -4, 9, -16, 25, …Squares of the positive integers, with alternating signs

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Identify the pattern in the sequence: the terms are squares of positive integers with alternating signs. The sequence is 1, -4, 9, -16, 25, ... which corresponds to 1^2, -2^2, 3^2, -4^2, 5^2, ...

Express the nth term as the square of n, which is \(n^2\).

Incorporate the alternating sign. Since the signs alternate starting with positive for n=1, use \((-1)^{n+1}\) to represent the sign pattern: positive when n is odd, negative when n is even.

Combine the sign and the square to write the general formula for the nth term as \(a_n = (-1)^{n+1} \times n^2\).

Verify the formula by plugging in the first few values of n to ensure it matches the given sequence.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sequences and Terms

A sequence is an ordered list of numbers where each number is called a term. The nth term represents the value at position n in the sequence. Understanding how to express the nth term as a formula allows you to find any term without listing all previous terms.

Square Numbers

Square numbers are integers raised to the power of two, expressed as n². In this sequence, each term’s absolute value corresponds to the square of its position index, such as 1², 2², 3², and so on, which helps identify the pattern in the sequence.

Alternating Signs

Alternating signs mean the terms switch between positive and negative values in a regular pattern. This can be represented mathematically using (-1)ⁿ or (-1)ⁿ⁺¹, which changes the sign of each term depending on whether n is even or odd.

Finding a Sequence’s FormulaIn Exercises 13–30, find a formula for the nth term of the sequence.1, -4, 9, -16, 25, …Squares of the positive integers, with alternating signs