Problem 1 - Algebra and Trigonometry

Question

Find the first four terms and the 100th term of the sequence. \(a_{n}=n+1\)

Step-by-Step Solution

Answer

The first four terms are 2, 3, 4, 5, and the 100th term is 101.

Step 1: Understand the Given Sequence Formula

The sequence is defined by the formula \(a_{n} = n + 1\). This means each term is generated by taking its position number \(n\) and adding 1.

Step 2: Calculate the First Term

To find the first term of the sequence, substitute \(n = 1\) into the formula: \[ a_{1} = 1 + 1 = 2\] So, the first term is 2.

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the formula to find the second term: \[ a_{2} = 2 + 1 = 3\] Thus, the second term is 3.

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the formula to find the third term: \[ a_{3} = 3 + 1 = 4\] So, the third term is 4.

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the formula to find the fourth term: \[ a_{4} = 4 + 1 = 5\] Therefore, the fourth term is 5.

Step 6: Calculate the 100th Term

To find the 100th term of the sequence, substitute \(n = 100\) into the formula: \[ a_{100} = 100 + 1 = 101\] So, the 100th term is 101.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term after the first is derived from the previous one by adding a constant difference. This type of sequence is straightforward and predictable. Understanding this concept helps in identifying patterns that repeat systematically throughout the sequence. For example:

• If you start with a number, say 2, and add 3 every time, you get an arithmetic sequence: 2, 5, 8, 11, ...
• The constant difference here is 3.
• Similarly, in the formula given, \(a_n = n + 1\), every new position just adds 1, hence the difference is 1.

Recognizing the sequence type allows you to predict further values without calculating each individually.

Sequence Formula

The sequence formula is a mathematical expression that dictates how each term is calculated based on its position in the sequence. For the exercise provided, the formula is \(a_n = n + 1\). This formula tells us:

• Each term's value is simply 1 more than its position number \(n\).
• It's crucial because it simplifies finding any term in the sequence directly without listing all prior terms.
• Knowing the formula makes it easy to compute terms like the 100th because you only need to substitute and calculate.

By understanding this formula, you can efficiently determine any term in the sequence without hassle.

Term Position

The term position, often represented by \(n\), is a key aspect of sequences. It denotes the specific place a term holds in a sequence which is essential for calculation purposes. Considerations around term position include:

• It helps identify the exact term you need when applying the formula.
• In \(a_n = n + 1\), \(n\) tells us our place in the sequence, making calculation straightforward.
• If you know \(n\), finding \(a_n\) is just a simple plug-in job.

Being adept at recognizing and using the term position streamlines both understanding and working with sequences.

Sequence Calculation

Sequence calculation involves using the provided formula to find specific terms in a sequence. It encompasses steps like substitution and arithmetic manipulation. With our formula \(a_n = n + 1\), sequence calculation means:

• Substituting the position number into the formula to get the term.
• It's a methodical process, where for the 1st term \(n = 1\), resulting in \(a_1 = 2\).
• For the 100th term, replacing \(n = 100\) yields \(a_{100} = 101\).

Efficient sequence calculation comes down to understanding the formula and systematically applying it to determine any term requested, making potentially complex series manageable.