Problem 36 - Algebra 2

Question

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 4,16,64,256,1024, \dots $$

Step-by-Step Solution

Answer

The sequence will continue with 4096 and 16384. The explicit formula is \(a_n = 4^n\), and the recursive formula is \(a_n = 4 \cdot a_{n-1}\) with \(a_1 = 4\).

Step 1: Identify the pattern

Viewing the sequence, observe that each term is four times the previous term. That is, \(16 = 4 \times 4\), \(64 = 4 \times 16\), \(256 = 4 \times 64\) and so on.

Step 2: Predict the next two terms

Using the established pattern (each term being four times the prior term), the next term after \(1024\) would be \(4 \times 1024 = 4096\). The term following \(4096\) would then be \(4 \times 4096 = 16384\).

Step 3: Formulate the explicit formula

An explicit formula defines the \(n\)th term of a sequence in terms of \(n\). The explicit formula for this sequence is \(a_n = 4^n\). This formula works because the first term is \(4^1 = 4\), the second term is \(4^2 = 16\), and so forth.

Step 4: Formulate the recursive formula

A recursive formula defines the \(n\)th term of a sequence in terms of one or more previous terms (in this case, the \(n-1\)th term). The recursive formula for this sequence is \(a_n = 4 \cdot a_{n-1}\) with \(a_1 = 4\). This formula works because each term is four times the previous term.

Step 5: Identify type of each formula

As described, \(a_n = 4^n\) is an explicit formula, since it expresses the \(n\)th term directly in terms of \(n\). The formula \(a_n = 4 \cdot a_{n-1}\) is a recursive formula, since it expresses the \(n\)th term in terms of the previous term, rather than in terms of \(n\).

Key Concepts

Recursive Formula

A recursive formula is a fundamental concept in sequences and series. It defines each term of a sequence based on the preceding term(s). This kind of formula is great for understanding how terms relate within the sequence. In this sequence, the recursive formula is \( a_n = 4 \cdot a_{n-1} \) starting with \( a_1 = 4 \).
This indicates each new term is four times the previous term:

• Starts at \( a_1 = 4 \).
• To find \( a_2 \), multiply \( a_1 \) by 4, resulting in 16.
• Proceed similarly for subsequent terms.

Although recursive formulas are powerful for building sequences, they are less convenient for directly finding individual terms without calculating all preceding terms. This explains why both recursive and explicit formulas are valuable.

Explicit Formula

An explicit formula, unlike a recursive formula, provides a direct way to calculate any term in a sequence without needing to know its predecessor.
For geometric sequences like the one we've explored, the explicit formula is particularly handy. In this example, the explicit formula is \( a_n = 4^n \).
This translates to:

• To find the third term, plug in \( n = 3 \) to get \( a_3 = 4^3 = 64 \).
• The power of 4 directly corresponds to the term number, meaning easy calculation.

This formula is efficient, especially when you need to find a large\( n \)-th term without computing all previous terms. Just "plug and play" with \( n \).

Geometric Sequence

A geometric sequence is a specific type of sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our sequence, the common ratio is 4.
This leads to the sequence: 4, 16, 64, 256, 1024, and so forth.

• The first term is 4.
• The common ratio, multiplied with each preceding term, is also 4.

Geometric sequences can be quickly identified by their multiplicative pattern. Once you know the common ratio, you can easily determine any other term, becoming a straightforward calculation or verification task. This fundamental understanding supports both the recursive and explicit formulas discussed.