Problem 14 - Algebra and Trigonometry

Question

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}-0.3, a_{1}=-1.7 $$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are -1.7, -2.0, -2.3, -2.6, -2.9, -3.2.

Step 1 - Find the First Term

The first term \(a_{1}\) is already given in the problem, and it is -1.7.

Step 2 - Find the Second Term

Determine the next term (\(a_{2}\)) by substituting \(n=2\) in the formula to get \(a_{2} = a_{2-1} - 0.3 = a_{1} - 0.3 = -1.7 - 0.3 = -2.0\).

Step 3 - Find the Third Term

By substituting \(n=3\) in the formula, you can get the third term (\(a_{3}\)). Thus, \(a_{3} = a_{3-1} - 0.3 = a_{2} - 0.3 = -2.0 - 0.3 = -2.3\).

Step 4 - Find the Fourth Term

The same operation should be repeated for the fourth term (\(a_{4}\)), thus \(a_{4} = a_{4-1} - 0.3 = a_{3} - 0.3 = -2.3 - 0.3 = -2.6\).

Step 5 - Find the Fifth Term

Find the fifth term (\(a_{5}\)) by substituting \(n=5\) in the formula, thus \(a_{5} = a_{5-1} - 0.3 = a_{4} - 0.3 = -2.6 - 0.3 = -2.9\).

Step 6 - Find the Sixth Term

Finally, find the sixth term (\(a_{6}\)) by substituting \(n=6\) in the formula, thus \(a_{6} = a_{6-1} - 0.3 = a_{5} - 0.3 = -2.9 - 0.3 = -3.2\).

Key Concepts

Sequence Notation

Understanding sequence notation is essential for identifying and working with sequences in mathematics. Essentially, a sequence is a list of numbers in a specific order. In mathematics, we use specific notations to represent sequences to make it easier to identify them and to work with them.

Sequence notation uses the format \( a_n \), where \( n \) represents the position of a term within the sequence, and \( a \) denotes the value of the term at that position. We call \( n \) the index or the subscript. For example, let's say you have a sequence where the first term is \( a_1 \), the second term is \( a_2 \) and so on. This notation helps us quickly refer to the terms and their positions in a sequence without writing out the entire sequence.

In the given arithmetic sequence, \( a_{n}=a_{n-1}-0.3, a_{1}=-1.7 \), \( a_1 \) is the first term, and to find any subsequent term, \( a_n \) (where \( n \) is greater than 1), you use the previous term (\( a_{n-1} \)) and subtract 0.3. This notation systematically represents the relationship between consecutive terms of the sequence.

Arithmetic Sequence Formula

An arithmetic sequence is one where each term after the first is obtained by adding a constant, known as the common difference, to the previous term. The arithmetic sequence formula for finding the \( n \)th term is given by \({ a_{n} = a_{1} + (n-1)d }\), where:\

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• \\( a_{n} \)\ is the \( n \)th term of the sequence,\
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• \\( a_{1} \)\ is the first term of the sequence,\
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• \\( n \)\ is the term number,\
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• \\( d \)\ is the common difference between the terms.\
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\In simple terms, to find any term in an arithmetic sequence, start with the first term and then add the common difference \( d \) the appropriate number of times. For example, the provided exercise contains an arithmetic sequence with \( a_{1}=-1.7 \) and a common difference of \( -0.3 \) (since the sequence is decreasing). Using the formula, we systematically determine each subsequent term.

Arithmetic Sequence Terms

Terms in an arithmetic sequence can be identified using the first term and the common difference. In the sequence \( a_{n}=a_{n-1}-0.3, a_{1}=-1.7 \), we know that each term is 0.3 less than the previous term. Therefore, the terms have a regular and predictable pattern.

The first term \( a_{1} \) is the starting point for the sequence. In this example, it is \( -1.7 \) which serves as the foundation for all other terms. Each subsequent term is found by subtracting the common difference \( -0.3 \) from the previous term. By continuing this pattern, we generate the terms of the sequence. Thus, the sequence progresses as follows: \( -1.7, -2.0, -2.3, -2.6, -2.9, -3.2 \).

Understanding the structure of terms in an arithmetic sequence allows for easy computation of any term within the sequence and lays the groundwork for further concepts in advanced mathematics, such as series and mathematical induction.