Problem 14 - Algebra and Trigonometry

Question

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=12 \text { and } a_{n}=a_{n-1}+4 \text { for } n \geq 2 $$

Step-by-Step Solution

Answer

The first four terms of the sequence are 12, 16, 20, 24.

Step 1: Initial Term

The very first term is given already: \(a_{1}=12\). No calculations are needed for this step.

Step 2: Second Term Calculation

The second term \(a_{2}\) can be obtained by putting \(n=2\) into the recurrence formula, which results in \(a_{2}=a_{2-1}+4 = a_{1} + 4 = 12 + 4 = 16\).

Step 3: Third Term Calculation

The third term \(a_{3}\) can be obtained similarly by putting \(n=3\) into the recurrence formula. So, \(a_{3}=a_{3-1}+4=a_{2}+4 = 16 + 4 = 20\)

Step 4: Fourth Term Calculation

Put \(n=4\) into the recurrence formula to get the fourth term \(a_{4}\). So, \(a_{4}=a_{4-1}+4=a_{3}+4= 20 + 4 = 24\)

Key Concepts

Recursion Formula

A recursion formula is a way of defining the terms of a sequence using previous terms. Think of it as a chain, where each link connects to the one before. This means if you know one term, you can find the next one by applying the given rule or formula.
In our exercise, the recursion formula tells us how to move from one term to another. It's given by:

• \( a_{n} = a_{n-1} + 4 \) for \( n \geq 2 \)

This formula means that every term (after the first) is the previous term plus 4. This type of formula is handy when you need to calculate sequence terms one after another. It's like following a recipe—once you have the ingredients for the first step, you can move smoothly through to the end. Remember, the starting term is crucial since it determines all the terms that follow. In this sequence, the first term is \( a_{1} = 12 \). Without this first term, the recipe would not work.

Arithmetic Sequence

An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant number to the previous term. This specific number is called the common difference.
In our exercise, we have an arithmetic sequence thanks to the recursion formula that adds 4 to each term. Here, the common difference is:

• 4

This pattern is what makes it an arithmetic sequence. For example, starting from 12, the sequence becomes 12, 16, 20, 24, and so on.
This means the terms form a straight line when you plot them on a graph because each term is evenly spaced. This property makes arithmetic sequences predictable and easy to graph. Arithmetic sequences are everywhere – think of regular savings, even footstep distances, or marching band movements. Recognizing these sequences makes it easier to understand their behavior.

Sequence Terms

Sequence terms are the individual elements or numbers that make up a sequence. Each of these numbers is usually represented by \( a_{n} \), where \( n \) denotes the position of the term in the sequence.
For the sequence in our exercise:

• \( a_{1} = 12 \)
• \( a_{2} = 16 \)
• \( a_{3} = 20 \)
• \( a_{4} = 24 \)

These terms are calculated starting from the first term and using the recursion formula to get the subsequent terms. Sequence terms can be infinite or finite depending on the rules given. In many practical situations, you might only need the first few terms.
Understanding sequence terms is essential as it helps in identifying patterns which can aid in predictions or solving further mathematical problems. It's like piecing together a puzzle of numbers, where each term provides more insight into the overall picture.