Problem 37 - Algebra and Trigonometry

Question

A sequence is defined recursively. List the first five terms. $$a_{1}=-2 ; \quad a_{n}=n+a_{n-1}$$

Step-by-Step Solution

Answer

The first five terms are: \( -2, 0, 3, 7, 12 \).

Step 1: Understand the initial term

The first term of the sequence, \( a_1 \), is given as \( -2 \). This is our starting point.

Step 2: Use the recursive formula for the second term

To find the second term, use the formula \( a_n = n + a_{n-1} \) with \( n = 2 \). Thus, \( a_2 = 2 + a_1 = 2 + (-2) = 0 \).

Step 3: Calculate the third term

Now, use the recursive formula for the third term with \( n = 3 \). Thus, \( a_3 = 3 + a_2 = 3 + 0 = 3 \).

Step 4: Determine the fourth term

Use the formula again for the fourth term with \( n = 4 \). Thus, \( a_4 = 4 + a_3 = 4 + 3 = 7 \).

Step 5: Find the fifth term

Finally, use the formula for the fifth term with \( n = 5 \). Thus, \( a_5 = 5 + a_4 = 5 + 7 = 12 \).

Key Concepts

sequence definition

A sequence is an ordered list of numbers. Each number in the sequence is called a term. These terms follow a specific pattern or rule to determine their value.

Sequences can be finite or infinite. A finite sequence has a limited number of terms, while an infinite sequence goes on endlessly. Sequences are widely used in mathematics and can represent various patterns.

For example, in the given problem, we have a sequence defined by a specific initial term and a recursive formula.

recursion in sequences

Recursion is a concept where a term in a sequence is defined using previous terms. This creates a relationship, where each term depends on one or more preceding terms.

In the given exercise, the sequence is defined recursively as follows:

- The first term is explicitly given: \(a_{1} = -2\)
- Every subsequent term is determined by adding the current position (\(n\)) to the previous term (\(a_{n-1}\)). This can be written as: \(a_{n} = n + a_{n-1}\)

The recursive rule provides a straightforward way to compute each term based on the one before it. Here’s a quick summary of how it applies to our sequence:

- Start with \(a_{1} = -2\)
- \(a_{2} = 2 + a_{1} = 2 + (-2) = 0\)
- \(a_{3} = 3 + a_{2} = 3 + 0 = 3\)
- \(a_{4} = 4 + a_{3} = 4 + 3 = 7\)
- \(a_{5} = 5 + a_{4} = 5 + 7 = 12\)

initial term and formula

The initial term and the recursive formula are essential components to define a sequence.

1. **Initial Term**: This is the starting term of the sequence. It is often denoted as \(a_{1}\), and in our problem, it is given as \(-2\). This initial value sets the stage for calculating all other terms in the sequence.

2. **Recursive Formula**: This is the rule that relates each term to its preceding term(s). The formula in the problem is \(a_{n} = n + a_{n-1}\). It defines how each term is derived from the previous term.

Using these elements, we can find any term in the sequence by starting with the initial term and applying the recursive formula repeatedly. As demonstrated in the solution steps, we start with \(a_{1} = -2\) and use the formula to calculate each subsequent term.

Understanding the initial term and the recursive rule is crucial to analyzing and generating the sequence.