Problem 94 Write a recursion formula for ea... [FREE SOLUTION]

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Precalculus Functions and Graphs

Mark Dugopolski

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 11: Problem 94

Write a recursion formula for each sequence. $$t^{2}, t^{4}, t^{8}, t^{16}$$

Short Answer

Expert verified

The recursion formula is $ a_n = (a_{n-1})^2 $ with $ a_1 = t^2 $.

Step by step solution

Identify the pattern in the sequence

Observe the given sequence: \[ t^{2}, t^{4}, t^{8}, t^{16} \]. Each term in the sequence is a power of 2. Notice that each term is obtained by raising $t$ to a power which is double the power of the previous term.

Express the relation between consecutive terms

To find the recursion formula, express each term in the sequence using the previous term: \[ t^4 = (t^2)^2 \] \[ t^8 = (t^4)^2 \] \[ t^{16} = (t^8)^2 \] Thus, each term is the square of the previous term.

Generalize the recursion formula

From the observation in Step 2, we can generalize the recursion formula for the sequence. Let $ a_n $ represent the $n$-th term of the sequence. Then, the relationship between the $n$-th term and the $(n-1)$-th term can be written as: \[ a_n = (a_{n-1})^2 \] where $ a_1 = t^2 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Patterns

In math, a sequence is an ordered list of numbers that follow a specific pattern. Understanding sequence patterns helps us predict future terms in the sequence.

For example, in the sequence \[ t^2, t^4, t^8, t^{16} \], we see that each term involves raising a variable, $t$, to a power. The key pattern here is that the exponent of $t$ doubles with each step:

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The first term is $t^2$

The second term is $t^4$ (which is $(t^2)^2$)

The third term is $t^8$ (which is $(t^4)^2$)

The fourth term is $t^{16}$ (which is $(t^8)^2$)

By identifying sequence patterns, we can make sense of how sequences progress and use this information to develop recursive formulas. In this case, each term is derived by squaring the previous term.

Exponential Functions

Exponential functions are a vital concept in mathematics and are represented as $a^x$, where $a$ is any constant and $x$ is the exponent. They grow (or decay) at rates proportional to their current value, creating a rapid increase (or decrease) over time.

In our sequence \[ t^2, t^4, t^8, t^{16} \], we can see that the sequence is based on exponential growth: the exponent is doubling each time.

Understanding exponential functions can make it easier to grasp this pattern. Each term can be considered an exponential function where the base $t$ has an ever-increasing exponent following the sequence $2, 4, 8, 16$.

Generalization in Sequences

Generalization in sequences involves finding a formula that applies to any term within the sequence, not just the first few. This is where a recursion formula becomes invaluable.

For the sequence $ t^2, t^4, t^8, t^{16} $, we observed that each term is the square of the previous one.

We start by noting the first term, $ a_1 = t^2 $
We then express each term as a function of the previous term: $a_{n} = (a_{n-1})^2$

With this recursive formula, we can calculate any term in the sequence by repeatedly applying this relationship. This type of generalization is useful for efficient calculations and deeper insights into the behavior of sequences.

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91影视

Write a recursion formula for each sequence. $$t^{2}, t^{4}, t^{8}, t^{16}$$

Short Answer

Step by step solution

Identify the pattern in the sequence

Express the relation between consecutive terms

Generalize the recursion formula

Key Concepts

Sequence Patterns

Exponential Functions

Generalization in Sequences

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