Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=n+5 $$

Step-by-Step Solution

Answer

The first five terms are 6, 7, 8, 9, and 10.

Step 1: Understand the Sequence Formula

The sequence is defined as \( a_n = n + 5 \). This formula gives us the general term of the sequence based on the index \( n \). To find the specific terms, we will substitute the values of \( n = 1, 2, 3, 4, \) and \( 5 \) into the formula.

Step 2: Calculate the First Term

To find the first term, substitute \( n = 1 \) into the formula:\[ a_1 = 1 + 5 = 6. \]Thus, the first term is 6.

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula:\[ a_2 = 2 + 5 = 7. \]Therefore, the second term is 7.

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula:\[ a_3 = 3 + 5 = 8. \]So, the third term is 8.

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula:\[ a_4 = 4 + 5 = 9. \]Hence, the fourth term is 9.

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \) into the formula:\[ a_5 = 5 + 5 = 10. \]Thus, the fifth term is 10.

Key Concepts

Sequence Formula

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is known as the common difference. To find any term in an arithmetic sequence, we use a sequence formula. The general sequence formula for an arithmetic sequence can be written as follows:

• \( a_n = a_1 + (n-1) \cdot d \)

In this formula, \( a_n \) represents the nth term in the sequence, \( a_1 \) is the first term, \( d \) is the common difference between terms, and \( n \) is the position (index) of the term in the sequence. By applying this formula, you can easily determine any term in the sequence without having to list all preceding terms.

Terms Calculation

Terms calculation involves substituting the position of each term into the sequence formula to find the corresponding term value. In our exercise, the formula is given as \( a_n = n + 5 \). To calculate the terms:

• First term: Substitute \( n = 1 \), resulting in \( a_1 = 1 + 5 = 6 \).
• Second term: Substitute \( n = 2 \), resulting in \( a_2 = 2 + 5 = 7 \).
• Third term: Substitute \( n = 3 \), leading to \( a_3 = 3 + 5 = 8 \).
• Fourth term: Substitute \( n = 4 \), resulting in \( a_4 = 4 + 5 = 9 \).
• Fifth term: Substitute \( n = 5 \), leading to \( a_5 = 5 + 5 = 10 \).

This method allows us to quickly calculate multiple terms once the sequence formula is known.

Index Substitution

Index substitution is a crucial step in finding the specific terms of a sequence. It's the process of substituting specific values of the index \( n \) into the sequence formula to determine the sequence's term at that position. For example, in the sequence formula \( a_n = n + 5 \), trying different values of \( n \) effectively gives us each term:

• For \( n = 1 \), substituting gives \( a_1 = 1 + 5 = 6 \).
• For \( n = 2 \), substituting gives \( a_2 = 2 + 5 = 7 \).
• Continuing this process similarly helps us find further terms.

Through index substitution, we leap directly to the desired term's value without sequential addition or complex calculations.

Explicit Formula

An explicit formula in sequences is a direct way to find any term without knowing the previous term. The explicit formula gives a direct expression where the term can be calculated explicitly for any position in the sequence. In arithmetic sequences, the explicit formula is:

• \( a_n = a_1 + (n-1) \cdot d \)

Here, each term is directly calculated by its index involving no recursive process. In the specific problem you've been tackling, the explicit formula is \( a_n = n + 5 \), meaning no need to use the previous term to find the next. This approach is particularly useful because it allows you to find any term of the sequence quickly and efficiently, even if you're interested in a very large index.