Determining the Second Term of a Sequence: A Comprehensive Guide

The concept of sequences is fundamental in mathematics, often appearing in algebra and calculus. This article will delve into how to determine the second term of a sequence given its general term formula. We will explore three different scenarios and provide detailed explanations for each case.

Understanding the General Term of a Sequence

A sequence is a set of numbers arranged in a specific order. The general term of a sequence, denoted as ( a_n ), provides a formula to calculate the value of any term in the sequence based on its position. Let's explore how to find the second term of a sequence in three different scenarios.

Scenario A: ( a_n n^2 - 1 )

In this scenario, the general term of the sequence is given by the formula ( a_n n^2 - 1 ). This formula represents a quadratic sequence, where each term is derived by squaring the position of the term and then subtracting 1.

To find the second term, we substitute ( n 2 ) into the formula:

[ a_2 2^2 - 1 ]

Calculating the value:

[ a_2 4 - 1 3 ]

Therefore, the second term of the sequence is 3.

Scenario B: ( a_n frac{n-1}{n 1} )

In this scenario, the general term of the sequence is given by the formula ( a_n frac{n-1}{n 1} ). This formula represents a sequence where each term is the fraction of the position minus 1 over the position plus 1.

To find the second term, we substitute ( n 2 ) into the formula:

[ a_2 frac{2-1}{2 1} ]

Calculating the value:

[ a_2 frac{1}{3} ]

Therefore, the second term of the sequence is ( frac{1}{3} ).

Scenario C: ( a_n 1 )

In this scenario, the general term of the sequence is given by the constant formula ( a_n 1 ). This formula represents a sequence where every term is equal to 1, regardless of the position.

To find the second term, we simply note that:

[ a_2 1 ]

Therefore, the second term of the sequence is 1.

Conclusion

Understanding how to determine the second term of a sequence is crucial in algebra and mathematical problem-solving. The scenarios covered in this article showcase the diversity of sequence formulas and how to apply them to find specific terms. Whether the sequence is quadratic, fractional, or constant, the key is to correctly substitute the position number into the given formula.

For further study, consider exploring more complex sequence formulas and their applications in various mathematical contexts. This knowledge will enhance your problem-solving skills and ability to understand the underlying mathematical principles.