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Chapter 11: Problem 5

Find the first term and the common difference. $$8,13,18,23, \dots$$

Short Answer

Expert verified
The first term is 8 and the common difference is 5.

Step by step solution

01

- Identify the first term

The first term in an arithmetic sequence is the first number in the list. Here, the first term is 8.
02

- Find the common difference

The common difference in an arithmetic sequence is the difference between any two successive terms. To find it, subtract the first term from the second term: common difference = 13 - 8 = 5.

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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 crucial because it acts as the starting point. This term is simply the initial number in the sequence.

For instance, in the arithmetic sequence provided, 8, 13, 18, 23, ..., the first term is 8.

The notation for the first term is often referred to as a1. This helps when writing and solving formulas related to arithmetic sequences.
Common Difference
The common difference is the amount by which each term in an arithmetic sequence increases or decreases.

You can find the common difference by subtracting the first term from the second term, or alternatively, by subtracting any term from the term that follows it.

For the sequence 8, 13, 18, 23, ...:
  • We subtract the first term from the second term: 13 - 8 = 5.
  • Thus, the common difference (d) is 5.
The common difference plays a crucial role in determining subsequent terms in the sequence.
Successive Terms
Once you know the first term and the common difference, you can find all other terms in the sequence, also known as the successive terms.

For any term in an arithmetic sequence, you can use the formula:

\[ a_n = a_1 + (n - 1)d \]

  • an is the nth term
  • a1 is the first term
  • n is the term鈥檚 position in the sequence
  • d is the common difference.
Using our example with a1 = 8 and d = 5, let's find the 4th term:

\[ a_4 = 8 + (4 - 1) \times 5 = 8 + 15 = 23 \]

Indeed, the 4th term matches our sequence's 4th term.

In summary, understanding how to identify the first term and common difference allows you to determine any term in an arithmetic sequence efficiently and accurately.

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Most popular questions from this chapter

Multiply: \(\left(x^{2}+2 x y+y^{2}\right)\left(x^{2}+2 x y+y^{2}\right)^{2}(x+y)\)

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Under what circumstances is it possible for the 5 th term of a geometric sequence to be greater than the 4 th term but less than the 7 th term?

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