Problem 17 - College Algebra

Question

Write the first four terms of each sequence. $$a_{1}=4 \text { and } a_{n}=2 a_{n-1}+3 \text { for } n \geq 2$$

Step-by-Step Solution

Answer

The first four terms of the sequence are 4, 11, 25, 53

Step 1: Identify the first term

The first term of our sequence (denoted as \(a_1\)) is given in the problem as 4. Therefore, the first term of the sequence is 4.

Step 2: Calculate the second term

Using the recursive formula \(a_n=2 a_{n-1}+3\), where \(n=2\) and \(a_{n-1}=a_1\), we find that \(a_2 = 2a_1 + 3 = 2*4 + 3 = 11\). Therefore, the second term of the sequence is 11.

Step 3: Calculate the third term

Similarly, for the third term, we replace \(n\) with 3 and \(a_{n-1}\) with \(a_2\) in the formula. This gives \(a_3 = 2*a_2 + 3 = 2*11 + 3 = 25\). Therefore, the third term of the sequence is 25.

Step 4: Calculate the fourth term

For the fourth term, we replace \(n\) with 4 and \(a_{n-1}\) with \(a_3\) in the formula. This gives \(a_4 = 2*a_3 + 3 = 2*25 + 3 = 53\). Therefore, the fourth term of the sequence is 53.

Key Concepts

Algebraic Sequences

Understanding algebraic sequences is crucial for grasping many mathematical concepts. An algebraic sequence is a list of numbers in which each term after the first is created by following a rule. It's similar to following a recipe; once you know the initial ingredient (the first term) and the process to make the rest (the formula), you can produce as many terms of the sequence as needed. Algebraic sequences can be arithmetic, where you add a constant to get from one term to the next, or geometric, where you multiply by a constant. Recursive sequences, a type of algebraic sequence, are particularly interesting because they build upon previous terms. Imagine climbing a ladder where each step up is determined by where the previous step is; that's similar to how recursive sequences work.

Recursive Formula

A recursive formula is a way of defining the terms of a sequence using previous terms. It's a bit like a domino effect: once you knock the first one over (or know the first term), the rest will fall into place one after the other. Typically, a recursive formula will have two parts: the starting value or values (like the first term or the first couple of terms) and a rule for finding every term after that. This rule takes into consideration one or more of the preceding terms. For example, a simple recursive sequence might start with a number and the rule will tell you to add 5 to the previous term to get the next one. This can lead to a fascinating growth pattern that's fun to follow for as many terms as you like.

Sequence Terms Calculation

Calculating the terms of a sequence is a straightforward process once you are familiar with the recursive formula. For the exercise at hand, where the first term is given and the rest of the terms are calculated based on the previous one, it's like piecing together a puzzle. Starting with the known first term, apply the recursive rule—multiply the previous term by 2 and add 3 in our example—to find the next term. Repeat this process, and watch the sequence unfold. It's like following a trail of breadcrumbs: every new term leads you to the next, and each is essential in finding the term that comes after. By computing one term at a time, carefully and methodically, you can map out the entire sequence, no matter how many terms you need.