Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=2 n-1 $$

Step-by-Step Solution

Answer

The first five terms are 1, 3, 5, 7, 9.

Step 1: Understand the sequence formula

The given sequence formula is \(a_n = 2n - 1\). This means that the nth term of the sequence is found by substituting the value of \(n\) into the formula. Our task is to find the first five terms, corresponding to \(n=1\) through \(n=5\).

Step 2: Calculate the first term

To find the first term of the sequence, substitute \(n=1\) into the formula:\[ a_1 = 2(1) - 1 = 2 - 1 = 1 \]Thus, the first term \(a_1\) is 1.

Step 3: Calculate the second term

To find the second term of the sequence, substitute \(n=2\) into the formula:\[ a_2 = 2(2) - 1 = 4 - 1 = 3 \]Thus, the second term \(a_2\) is 3.

Step 4: Calculate the third term

To find the third term of the sequence, substitute \(n=3\) into the formula:\[ a_3 = 2(3) - 1 = 6 - 1 = 5 \]Thus, the third term \(a_3\) is 5.

Step 5: Calculate the fourth term

To find the fourth term of the sequence, substitute \(n=4\) into the formula:\[ a_4 = 2(4) - 1 = 8 - 1 = 7 \]Thus, the fourth term \(a_4\) is 7.

Step 6: Calculate the fifth term

To find the fifth term of the sequence, substitute \(n=5\) into the formula:\[ a_5 = 2(5) - 1 = 10 - 1 = 9 \]Thus, the fifth term \(a_5\) is 9.

Step 7: Summarize the first five terms

Based on the calculations, the first five terms of the sequence are: 1, 3, 5, 7, and 9.

Key Concepts

Sequence Formula

A sequence formula is like a recipe in mathematics. It tells us exactly how to find each term in a sequence. In the problem we explored, the sequence formula is given as \( a_n = 2n - 1 \). This formula determines the sequence's terms by taking an input number, \( n \), commonly called the 'term number', and using it in a calculation or rule.
The result from that formula gives us the value of each term. The sequence starts from \( n=1 \) and progresses sequentially. Thus, understanding and applying the formula correctly is key to discovering all the terms in the sequence.
Sequences are widely used to model various real-life situations, from arithmetic progressions to financial modeling.

Nth Term

The "nth term" is a term we often encounter when discussing sequences. It represents any general term in the sequence based on its position number, \( n \). By using the formula like \( a_n = 2n - 1 \), we can find the nth term for any position in the sequence.
This general term gives us tremendous power because once we have it, we can find any specific term without having to list all the preceding terms.

• To find the first term, put \( n = 1 \) in the formula, giving us 1.
• For the second term, substitute \( n = 2 \), which gives us 3.
• Continue this pattern sequentially to find as many terms as desired.

Exploring the nth term mathematically highlights patterns and offers a flexible method for finding sequence components.

Series Calculation

Series calculation involves summing up the terms of a sequence. Unlike a sequence which lists individual elements, a series combines them.
For an arithmetic sequence like the one we discussed, calculating a series might involve adding up a set number of terms. Here's how:

• Select the terms you want to include in the series, say the first five: 1, 3, 5, 7, and 9.
• Add them together: \( 1 + 3 + 5 + 7 + 9 \).
• The resulting sum is 25, which represents the series of the first five terms.

Series calculations can be extended to find sums over larger terms, using more advanced formulas, or software tools for ease and accuracy.

Identifying Patterns

Recognizing patterns in sequences helps us predict future terms and verify our mathematical calculations. In the sequence we're dealing with, \( a_n = 2n - 1 \), there's a visible pattern.
Each term increases by 2, showing a consistent difference, common in arithmetic sequences. Observing patterns allows for quicker predictions without recalculating every term.

• The first term is 1.
• The second term jumps to 3, increasing by 2.
• This increase continues uniformly: 5, 7, 9, ...

Recognizing such regularities aids in understanding complex sequences. It also illuminates how sequences evolve, simplifying series predictions and enhancing mental math skills.