Problem 11 - Algebra and Trigonometry

Question

Write the first six terms of each arithmetic sequence. $$ a_{n}=a_{n-1}-10, a_{1}=30 $$

Step-by-Step Solution

Answer

The first six terms of the arithmetic sequence are 30, 20, 10, 0, -10, -20.

Step 1: Identify the first term

The first term of the sequence, \(a_{1}\), is given as 30.

Step 2: Apply the recursive formula for the second term

Using the formula \(a_{n}=a_{n-1}-10\), where \(n=2\) and \(a_{n-1}=30\), the second term can be determined. The calculation will be as follows: \(a_{2}= a_{2-1}-10 = 30-10 = 20\).

Step 3: Apply the recursive formula for the third term

Repeat this approach for \(n=3\) to find \(a_{3}\). Use \(a_{2}=20\) in the recursive formula: \(a_{3}= a_{2}-10 = 20-10 = 10\).

Step 4: Apply the recursive formula for the fourth term

For \(n=4\), repeated application of the recursive formula gives \(a_{4}= a_{3}-10 = 10-10 = 0\).

Step 5: Apply the recursive formula for the fifth and sixth terms

Continuing in the same manner, it's possible to compute the next terms \(a_{5}=a_{4}-10 = 0-10 =-10\) and \(a_{6}=a_{5}-10 = -10-10 =-20\).

Step 6: Write down the first six terms of the sequence

By following the steps above, the first six terms of the arithmetic sequence are 30, 20, 10, 0, -10, -20.

Key Concepts

Recursive Formula

In mathematics, a recursive formula is a rule or equation that defines each term of a sequence using the previous term(s). This formula is particularly useful for sequences where each term builds directly on the one before. In the context of an arithmetic sequence, the recursive formula takes a simple form, often involving an addition or subtraction of a constant.
For this specific exercise, the recursive formula is given as follows:

• \( a_{n} = a_{n-1} - 10 \)

This means that to find any term in the sequence, you simply take the previous term and subtract 10. This rule allows us to calculate each subsequent term based on the current term, making it incredibly efficient for both writing and calculating sequence terms. Recursive formulas are powerful in simplifying calculations, especially when dealing with long sequences.

Sequence Terms

Sequence terms refer to the individual values or elements that make up a sequence. In the given arithmetic sequence, the terms are determined using the recursive formula: \( a_{n} = a_{n-1} - 10 \).
Initially, we know our first term \( a_{1} \) is 30. From there, each term is calculated as follows:

• The second term \( a_{2} \) is 20, since \( a_{2} = 30 - 10 \).
• The third term \( a_{3} \) is 10, deriving from \( a_{3} = 20 - 10 \).
• The fourth term \( a_{4} \) is 0, calculated as \( a_{4} = 10 - 10 \).
• The fifth term \( a_{5} \) is -10, found via \( a_{5} = 0 - 10 \).
• The sixth term \( a_{6} \) is -20, given by \( a_{6} = -10 - 10 \).

These are the first six terms of our sequence—each calculated by applying the recursive formula to its predecessor. Sequence terms in arithmetic sequences can be easily predicted and calculated once you understand the pattern established by the common difference.

Common Difference

The common difference in an arithmetic sequence is the constant amount you add or subtract during each step from one term to the next. It is what differentiates arithmetic sequences from geometric ones. This difference is pivotal for understanding the pattern of the sequence and simplifying the calculations involved.
In this problem, the common difference is -10, as specified in the recursive formula \( a_{n} = a_{n-1} - 10 \).
Identifying the common difference helps:

• Provide insight into how the sequence develops over time.
• Predict future terms beyond what's immediately calculated.
• Simplify the structuring of the sequence when writing it out fully.

Once you know the common difference, constructing an arithmetic sequence becomes straightforward, as each term is found simply by adjusting the previous term by this constant amount.