Problem 25 - Contemporary Precalculus

Question

Find the first five terms of the recursively defined sequence. $$a_{1}=4 \text { and } a_{n}=2 a_{n-1}+3 \text { for } n \geq 2$$

Step-by-Step Solution

Answer

Question: Find the first five terms of the recursively defined sequence given by: $$a_{1} = 4 \text { and } a_{n} = 2a_{n-1} + 3 \text { for } n \geq 2$$ Answer: The first five terms of the sequence are \(4, 11, 25, 53, \text { and } 109\).

Step 1: Find the first term

The first term is given as \(a_1 = 4\).

Step 2: Find the second term

Use the recursive formula \(a_n = 2a_{n-1} + 3\) with \(n=2\): $$a_2 = 2a_{2 - 1} + 3 = 2a_1 + 3 = 2 \cdot 4 + 3 = 11$$

Step 3: Find the third term

Use the recursive formula \(a_n = 2a_{n-1} + 3\) with \(n=3\): $$a_3 = 2a_{3 - 1} + 3 = 2a_2 + 3 = 2 \cdot 11 + 3 = 25$$

Step 4: Find the fourth term

Use the recursive formula \(a_n = 2a_{n-1} + 3\) with \(n=4\): $$a_4 = 2a_{4 - 1} + 3 = 2a_3 + 3 = 2 \cdot 25 + 3 = 53$$

Step 5: Find the fifth term

Use the recursive formula \(a_n = 2a_{n-1} + 3\) with \(n=5\): $$a_5 = 2a_{5 - 1} + 3 = 2a_4 + 3 = 2 \cdot 53 + 3 = 109$$ The first five terms of the sequence are \(4, 11, 25, 53, \text { and } 109\).

Key Concepts

Recursively Defined Sequence

A recursively defined sequence is a sequence in which each term after the first is defined as a function of the preceding terms. Unlike explicit formulae where we can find any term directly, a recursive formula provides us with the building blocks to derive one term after the other in succession.

Sequence and Series

In precalculus, 'sequence' refers to an ordered list of numbers following a certain pattern, while 'series' is the sum of the elements of a sequence. Understanding sequences is vital because they form the foundation for series, which are essential in various areas of mathematics, including calculus. Recursive sequences are just one type of sequence students might encounter, next to others like arithmetic or geometric sequences.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is created by adding a constant, called the common difference, to the previous term. This is distinct from a recursively defined sequence where the relationship can be more complicated. For example, an arithmetic sequence could start with 3 and have a common difference of 5, giving us the sequence 3, 8, 13, 18, and so on. The explicit formula for the n-th term of an arithmetic sequence is given by: \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.