Nth Term of Linear and Quadratic Sequences

1. Understanding and Finding the Nth Term of Linear Sequences

A linear sequence (also called an arithmetic progression) is a list of numbers that increases or decreases by the same amount each time. This constant amount is called the common difference.

Using the nth term:
The "nth term" is a formula that allows you to find any term in the sequence without having to list all the terms before it. For example, if you know the nth term, you can find the 100th term instantly by substituting $n=100$.

Finding the nth term:
There is a quick and reliable method for finding the nth term of any linear sequence.

Let's summarise that two-step method:

1. Find the common difference, d. This value will be the coefficient of n in your formula. For example, if the sequence goes up in 4s, the formula starts with $4n$. If it goes down in 7s, it starts with $-7n$.
2. Find the 'zero-th term'. Imagine the term that would come before the first term. To find it, take the first term and do the opposite of the common difference. This value is the constant you add or subtract at the end.

Example: Find the nth term of the sequence 5, 8, 11, 14, ...

1. The common difference is +3. So, the formula starts with $3n$.
2. The first term is 5. The "zero-th term" would be $5-3=2$.
3. So, the nth term is **$3n+2$
   **.

Let's watch this method in action.

2. Finding the Nth Term of Quadratic Sequences

Quadratic sequences are a step up in complexity. The difference between terms is not constant, but the second difference (the difference between the differences) is constant. The nth term will always be in the form **$a{n}^2+bn+c$
**.

Finding this formula requires a systematic approach. Messy working can easily lead to the arithmetic errors you're working to eliminate, so a clear layout is key.

Here is a visual summary of that process:

Let's watch a final video to consolidate this method.

3. Avoiding Common Errors (The Error-Loop Method)

As part of our course's "Error-Loop Method," it's vital to identify common pitfalls before you fall into them. For quadratic sequences, most errors are simple arithmetic slips that have big consequences.

Conclusion

Today you've learned the systematic methods for finding the rules behind both linear and quadratic sequences. These are fundamental skills that not only appear as standalone questions but also form the basis for more complex problems.

Key Takeaways:

• Linear Sequences: Identify the common difference (d) to get dn, then adjust with the "zero-th term".
• Quadratic Sequences: The process is rigid and must be followed carefully:
  1. Find the constant second difference.
  2. Halve it to find the coefficient of $n^2$ (the a value).
  3. Subtract the $a{n}^2$ sequence from your original sequence.
  4. Find the nth term of the resulting linear sequence (bn+c).
  5. Combine the parts to get $a{n}^2+bn+c$.
• Methodical Working: Neat, tabular working is your best defence against small calculation errors, especially in quadratic sequence questions.

For extra practice, you can work through the questions at the end of the Nth Term Notes PDF and the Quadratic Sequences webpage.

Preview of the Next Lesson:
We will continue our exploration of Module 5 by moving on to function notation. You will learn how to read and use notation like $f(x)$, evaluate composite functions like $fg(x)$, and find the inverse of a function, $f^{-1}(x)$.