Question

Find the next three terms of each sequence. $$2,-3,-8,-13, \ldots$$

Short answer

The next three terms are -18, -23, and -28.

Step-by-step solution

Identify the Pattern

Examine the differences between consecutive terms in the sequence: \(-3 - 2 = -5\), \(-8 - (-3) = -5\), \(-13 - (-8) = -5\). The differences are consistent, suggesting this is an arithmetic sequence with a common difference of \(-5\).

Use the Arithmetic Sequence Formula

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. We already have \(a_1 = 2\) and \(d = -5\).

Calculate the Next Term

To find the next term in the sequence (5th term), use the formula: \(a_5 = 2 + (5-1)(-5)\). Simplifying gives \(a_5 = 2 - 20 = -18\).

Calculate the Sixth Term

Proceed to the sixth term: \(a_6 = 2 + (6-1)(-5)\). Simplifying gives \(a_6 = 2 - 25 = -23\).

Calculate the Seventh Term

Finally, calculate the seventh term: \(a_7 = 2 + (7-1)(-5)\). Simplifying gives \(a_7 = 2 - 30 = -28\).

Key concepts

Understanding Common Difference

In arithmetic sequences, a common difference is crucial. It is simply the difference between consecutive terms. For example, in the given sequence \(2, -3, -8, -13, \ldots\), if we look closely at the terms:

• -3 - 2 = -5
• -8 - (-3) = -5
• -13 - (-8) = -5

This shows that the common difference is \(-5\). A negative common difference means we're subtracting each time which makes the sequence decrease. Conversely, a positive common difference would result in an increasing sequence. Recognizing this pattern is key to understanding how each term in the sequence is related to its neighbors.

Recognizing the Sequence Pattern

A sequence pattern in arithmetic sequences is determined by the repetition of adding (or subtracting) the common difference. Once you know the common difference, you can predict all future terms in the sequence. Observing the same sequence again: \(2, -3, -8, -13, \ldots\), the pattern is clear: each term is obtained by subtracting 5 from the previous one.This orderly arrangement of numbers, where each term is adjusted by a fixed amount, is what we call a sequence pattern. Patterns like these help in visualizing and predicting the behavior of the sequence. Understanding this not only aids in solving specific problems but also strengthens problem-solving skills related to sequences.

Applying the n-th Term Formula

The \(n\)-th term formula is a handy tool for finding any term in an arithmetic sequence without listing all previous numbers. It is given by:\[ a_n = a_1 + (n-1) \cdot d \],where \(a_n\) is the \(n\)-th term, \(a_1\) is the first term, and \(d\) is the common difference.In our sequence, \(a_1 = 2\) and \(d = -5\). To find, say, the 5th term \(a_5\):

• Substitute into the formula: \(a_5 = 2 + (5-1)(-5)\)
• Calculate: \(a_5 = 2 - 20 = -18\)

This calculation can be repeated to find any term, like the 6th or 7th term. The \(n\)-th term formula thus simplifies the process, making it efficient and less prone to error compared to manually calculating each step. It shows how algebra can transform repetitive calculations into simple, elegant expressions.