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Finite Differences

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Finite Differences

Study Guide

Key Definition

Finite differences are found by calculating the differences between consecutive terms in a sequence. For example, for the quadratic sequence $a_{n} = n^2$, the first difference is $a_{n} - a_{n-1} = 2n+1$.

Important Notes

Finite differences are used to identify patterns in sequences.
For quadratic sequences, forward differences form an arithmetic sequence.
Understanding finite differences helps in predicting sequence behavior.
Sequence of differences can reveal the degree of the polynomial.
Useful in calculus for approximating derivatives.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

$\sqrt{x}$ represents the square root of $x$

$Δa_n = a_n - a_{n-1}$ represents the finite difference

Remember to use proper notation when solving problems

Why It Works

Finite differences help in identifying the type of polynomial function generating a sequence by observing the pattern of differences.

Remember

The first forward difference of a quadratic sequence is linear, and the second forward difference is constant.

Quick Reference

Quadratic Sequence Formula:$a_{n} = n^2$

Forward Differences:$a_{n} - a_{n-1} = 2n+1$

Understanding Finite Differences

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Finite differences involve subtracting adjacent terms in a sequence, like $4 - 1 = 3$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the finite difference between the terms $4$ and $9$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You notice a pattern in your savings account balance over the months: $100, 150, 210, 280, \ldots$. Calculate the first finite difference.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a sequence: $1, 4, 9, 16, 25, 36, 49, \ldots$. What is the second finite difference?

Challenge Quiz

Single Choice Quiz

Advanced

For a sequence generated by $a_{n} = n^3$, what is the third finite difference?

Recap

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Review key concepts and takeaways