Problem 36 - Algebra 2

Question

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms. $$ 1,4,9,16, \dots $$

Step-by-Step Solution

Answer

This sequence is neither arithmetic nor geometric, it's a sequence of squares. The next two terms are 25 and 36.

Step 1: Check if It's an Arithmetic Sequence

An arithmetic sequence has a constant difference between consecutive terms. Calculating the differences, we have \(4 - 1 = 3\), \(9 - 4 = 5\), and \(16 - 9 = 7\). Since the differences vary, this isn't an arithmetic sequence.

Step 2: Check if It's a Geometric Sequence

A geometric sequence has a common ratio between consecutive terms. However, this sequence clearly shows addition of rising numbers, not multiplication, so it's not a geometric sequence.

Step 3: Identify the Sequence

Since the sequence isn't arithmetic or geometric, it's considered to be 'neither.' Observing the values, they correspond to the squares of the integer numbers 1, 2, 3, 4, ... i.e., \(1^2, 2^2, 3^2, 4^2, ...\). This pattern is likely to continue.

Step 4: Find the Next Two Terms

Following the identified pattern of square numbers, the next term corresponds to \(5^2\), which equals 25. The term after that corresponds to \(6^2\), equaling 36.

Key Concepts

Understanding Arithmetic Sequences

Arithmetic sequences are sequences of numbers in which the difference between consecutive terms is constant. This means if you subtract the first term from the second, the second from the third, and so on, you would always get the same result. This constant difference is called the 'common difference'.
For instance:

• An example sequence could be 2, 5, 8, 11, ...
• Here, the difference between terms is 3 (i.e., 5 - 2 = 3, 8 - 5 = 3).
• Thus, 3 is the common difference.

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

• \( a_n = a_1 + (n - 1) imes d \)
• where \( a_n \) is the nth term, \( a_1 \) is the first term, \( n \) is the number of terms, and \( d \) is the common difference.

Identifying an arithmetic sequence is all about finding this common difference. If it varies, then the sequence is likely not arithmetic.

Exploring Geometric Sequences

Geometric sequences are different from arithmetic sequences in a crucial way. In a geometric sequence, each term is derived by multiplying the previous one by a fixed, non-zero number known as the 'common ratio'.
Take a look at this example:

• A sequence like 3, 6, 12, 24, ...
• In this case, each term is multiplied by 2 to get the next one (i.e., 6 = 3 × 2, 12 = 6 × 2).
• Therefore, 2 is the common ratio.

You can determine any term in a geometric sequence using the formula:

• \( a_n = a_1 imes r^{(n-1)} \)
• where \( a_n \) is the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Identifying if a sequence is geometric involves checking if a consistent multiplication pattern exists through the entire sequence. If there is none, it isn't likely to be a geometric sequence.

Discovering Square Numbers

Square numbers form a fascinating sequence derived from squaring integers. Essentially, each term in this sequence corresponds to an integer raised to the power of two.
This is how it looks:

• The sequence 1, 4, 9, 16, ...
• is derived from squaring the numbers 1, 2, 3, 4, ... respectively (i.e., \(1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16\)).

To find any term in a sequence of square numbers, the formula is simple:

• \( a_n = n^2 \)
• where \( a_n \) is the nth term and \( n \) is the integer being squared.

Square number sequences are not classified as either arithmetic or geometric because their method of calculation is unique to the squaring of each integer. It leads to a familiar yet different progression in mathematics compared to other forms of sequences.