Question

Describe the pattern in each sequence, and use the pattern to find the next three terms. $$a, c, e, g, i, \ldots$$

Short answer

The next three terms are 'k', 'm', 'o'.

Step-by-step solution

Identify the Pattern

First, observe the given sequence: \(a, c, e, g, i, \ldots\). Notice the position of each letter in the alphabet: a (1st), c (3rd), e (5th), g (7th), i (9th). This sequence appears to be moving up the alphabet by skipping one letter between each term.

Establish the Pattern

From Step 1, determine the step pattern: each letter is the next odd-numbered letter in the alphabet. Specifically, it is moving through every second letter starting from 'a'.

Apply the Pattern to Find the Next Terms

Continuing with the pattern of every second letter gives the next terms: the letter that follows 'i' skipping one letter is 'k' (11th), then 'm' (13th), and 'o' (15th).

Verify the Solution

To confirm, check the alphabet positions: 'i' is 9th, 'k' is 11th, 'm' is 13th, and 'o' is 15th, which follows our observed pattern.

Key concepts

Alphabet Sequence

Understanding sequences that use the alphabet is incredibly useful in pattern identification. In our sequence, we start with the letter 'a', followed by 'c', 'e', 'g', and 'i'. To make sense of this, remember that each letter in the alphabet corresponds to a number position.

For example:

• 'a' is the 1st letter
• 'c' is the 3rd letter
• 'e' is the 5th letter
• 'g' is the 7th letter
• 'i' is the 9th letter

By understanding these numerical positions, it becomes clearer how each letter follows a specific pattern within the alphabet.

Step Pattern

When dealing with sequences, identifying the step pattern is vital. In our example sequence, the step pattern shows how each letter appears at specific intervals.

In this case, each letter is two positions away from the previous one in the list. This means we're skipping one letter in between each term. To confirm, look at the jump between the positions of letters:

• 1 to 3 (a to c)
• 3 to 5 (c to e)
• 5 to 7 (e to g)
• 7 to 9 (g to i)

Recognizing this pattern helps us predict future elements in the sequence through simple arithmetic.

Odd-Number Positions

In this sequence, each letter is located at an odd-numbered position in the alphabet. The letters are 1st (a), 3rd (c), 5th (e), 7th (g), and 9th (i). To find the next terms, look for the next odd positions.

The next odd positions after 9 are:

• 11th (k)
• 13th (m)
• 15th (o)

Thus, the continuation of our sequence is 'k', 'm', and 'o'.

This concept underscores the importance of recognizing how numbers and positions relate to patterns, especially when dealing with ordered sets like the alphabet.