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Chapter 10: Problem 14

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

Short Answer

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

Step by step solution

01

Identify first term and common difference

In this arithmetic sequence, the first term \(a_{1}\) is -1.7 and the common difference (known because each term is subtracted 0.3 from the previous one) is -0.3.
02

Calculate second term

The second term \(a_{2}\) would be the first term -0.3, which is -1.7 - 0.3 therefore \(a_{2}=-2.0\).
03

Calculate third term

Following the pattern, the third term \(a_{3}\) would be the second term - 0.3. So, \(a_{3}\) is -2.0 - 0.3 = -2.3.
04

Calculate fourth term

The fourth term \(a_{4}\) is the third term - 0.3, so \(a_{4}\) is -2.3 - 0.3 = -2.6.
05

Calculate fifth term

The fifth term \(a_{5}\) is the fourth term - 0.3, so \(a_{5}\) is -2.6 - 0.3 = -2.9.
06

Calculate sixth term

Finally, the sixth term \(a_{6}\) is the fifth term - 0.3, so \(a_{6}\) is -2.9 - 0.3 = -3.2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Term
In an arithmetic sequence, the first term is vital as it serves as the starting point. It is the initial value from which all subsequent terms are derived by following a specific pattern. In this exercise, the first term, denoted as \( a_1 \), is given as -1.7.
This value sets the stage for the entire sequence and helps to establish the progression of numbers. Understanding the first term is like building a foundation.
It is the number from which all calculations for future terms will begin. Without knowing the first term, it would be impossible to construct the sequence accurately.Remember, in any arithmetic sequence, identifying the first term helps you determine the entire sequence's trajectory.
Common Difference
The common difference in an arithmetic sequence is the key factor that determines the constancy of change between consecutive terms. This is the number that you repeatedly add to (or subtract from) the previous term to derive the next one.
For the given sequence, the common difference is -0.3. This tells us that each term is obtained by subtracting 0.3 from the previous term. Understanding the common difference is crucial as:
  • It dictates the gap between consecutive numbers in the sequence.
  • It can be positive or negative, affecting whether the sequence increases or decreases.
  • Knowledge of the common difference makes generating consecutive terms straightforward.
Once you have the common difference, you are equipped to continue extending the sequence indefinitely using the recursive formula.
Recursive Formula
The recursive formula is a way to define the terms of a sequence with reference to the preceding terms. It provides a method to describe each term based on its predecessor.
For the arithmetic sequence at hand, the recursive formula is given by \( a_n = a_{n-1} - 0.3 \). Here, \( a_n \) signifies the nth term, while \( a_{n-1} \) is the preceding term.This formula is essential for several reasons:
  • It is easy to use once the first term is known.
  • You only need the previous term to find the next, simplifying calculations.
  • It captures the entire process of generating a sequence compactly and accurately.
Using the recursive formula, you can efficiently continue the sequence beyond the explicit terms given, following the same pattern seamlessly.

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