what is the first of this fibonacci sequence: blank, -5/4, -2, -13/4

To solve for the first term in the Fibonacci sequence where the known terms are:

\[
x, -\frac{5}{4}, -2, -\frac{13}{4}
\]

we recall that in a Fibonacci sequence, each term is the sum of the two preceding ones. Therefore:

1. \(-2 = -\frac{5}{4} + t_1\)
2. \(-\frac{13}{4} = -2 + \left(-\frac{5}{4}\right)\)

Let’s solve these step by step:

### Step 1: Find the first term using the second equation
\[
-\frac{13}{4} = -2 - \frac{5}{4}
\]
Rewrite \(-2\) as \(\frac{-8}{4}\):
\[
-\frac{13}{4} = \frac{-8}{4} - \frac{5}{4}
\]
\[
-\frac{13}{4} = \frac{-13}{4}
\]
This confirms that the second equation holds true.

### Step 2: Solve for \(t_1\)
Now, solve the first equation:
\[
-2 = -\frac{5}{4} + t_1
\]
Add \(\frac{5}{4}\) to both sides:
\[
t_1 = -2 + \frac{5}{4}
\]
Rewrite \(-2\) as \(\frac{-8}{4}\):
\[
t_1 = \frac{-8}{4} + \frac{5}{4}
\]
\[
t_1 = \frac{-3}{4}
\]

So, the first term of the sequence is:

\[
t_1 = \frac{-3}{4}
\]

Would you like a detailed breakdown or have any other questions? 

Here are 5 related questions to explore:

1. How do Fibonacci sequences relate to the golden ratio?
2. What happens to a Fibonacci sequence if the first two terms are both negative?
3. How can Fibonacci sequences be generalized to complex numbers?
4. How is the Fibonacci sequence applied in nature or art?
5. What are Lucas sequences, and how do they differ from Fibonacci sequences?

**Tip:** In any recursive sequence like the Fibonacci, understanding the relationship between terms helps in constructing or deconstructing the sequence efficiently!

What is the first of this Fibonacci sequence: blank, -5/4, -2, -13/4

Discover how to find the first term in a Fibonacci sequence where the known terms include negative fractions. This problem demonstrates the use of recursive sequences and algebraic manipulation. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation