Question

Describe the pattern in each sequence, and use the pattern to find the next three terms. a, c, e, g, __, __, __

Short answer

The next three terms are i, k, m.

Step-by-step solution

Identify the Sequence Pattern

Look at the given sequence: a, c, e, g. Notice that each term is a letter in the English alphabet.

Determine the Increment

Find the alphabetical order of each letter: 'a' is 1, 'c' is 3, 'e' is 5, and 'g' is 7. The pattern shows that each letter is 2 positions after the previous one.

Apply the Pattern to Find the Next Term

Starting from 'g', add 2 to the position to get the next letter. 'g' (7) + 2 = 9, which is 'i' in the alphabet.

Find the Following Term

From 'i', add 2 to the position to get the next letter. 'i' (9) + 2 = 11, which is 'k' in the alphabet.

Find the Final Term

From 'k', add 2 to the position to get the next letter. 'k' (11) + 2 = 13, which is 'm' in the alphabet.

Key concepts

Sequence Pattern Recognition

In this exercise, we need to recognize the pattern within a sequence of letters: a, c, e, g.

Sequence pattern recognition involves analyzing sequences to determine a consistent rule or formula.
By identifying this rule, you can predict subsequent elements.

The first thing to notice is the sequence consists of letters from the English alphabet.
By focusing on the positions of these letters, you can uncover the hidden pattern.

Alphabet Increments

Examining the given sequence: a, c, e, g, we see each letter appears a few steps ahead of the previous one.

To decode the pattern, assign numerical positions to each letter.
- 'a' is in position 1
- 'c' is in position 3
- 'e' is in position 5
- 'g' is in position 7

Notice how the positions increase by 2 each time. This indicates the sequence follows an increment of 2.

Problem-Solving Steps

Solving pattern recognition problems involves systematic steps:

• Identify the pattern
• Determine the increment
• Apply the rule to find subsequent terms

Starting with our identified pattern and increment, we apply the solution step-by-step:
1. From 'g' (position 7), add 2 to reach 'i' (position 9)
2. From 'i' (position 9), add 2 to reach 'k' (position 11)
3. From 'k' (position 11), add 2 to reach 'm' (position 13)

Mathematical Sequences

Mathematical sequences often involve numbers, but they can also involve letters.

For letter sequences:
- Convert letters to their corresponding numerical positions in the alphabet.
- Identify increments or patterns in these numerical positions.
The sequence can then be considered an arithmetic sequence with a common difference.

For our example sequence:
- The common difference is 2.
This turns our letter sequence problem into a simple arithmetic sequence problem with terms like 1, 3, 5, 7, etc.
By mathematical rules, continuing the sequence is straightforward: 1+2 = 3, 3+2 = 5, 5+2 = 7, and so on.