The nth Term of a Sequence Revision Notes for Leaving Cert Mathematics

The nth Term of a Sequence

What is the nth term?

When working with number sequences, we often need to find a specific term without writing out all the previous terms. For example, if we want the 50th or 100th term of a sequence, writing out all those terms would be very time-consuming.

The nth term is a rule or formula that allows us to find any term in a sequence directly. This rule is usually written as $T_n$ where $n$ represents the position of the term we want to find.

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The power of the nth term formula becomes clear when working with large sequences. Instead of calculating 99 previous terms to find the 100th term, you can use the formula to calculate it directly in one step!

Key terminology

$T_n$ : The nth term formula
$n$ : The position number of the term (1st, 2nd, 3rd, etc.)
Common difference: The same amount added between consecutive terms in an arithmetic sequence

Understanding how the nth term works

If we have the rule $T_n = 2n + 3$ , we can find any term by substituting the position number:

$T_1 = 2(1) + 3 = 5$ (1st term)
$T_2 = 2(2) + 3 = 7$ (2nd term)
$T_3 = 2(3) + 3 = 9$ (3rd term)

This generates the sequence: 5, 7, 9, 11, 13, ...

The rule $T_n = 2n + 3$ allows us to find any term in this sequence without calculating all the previous terms.

Finding the nth term of an arithmetic sequence

For arithmetic sequences (sequences with a constant difference between terms), we can find the nth term using this systematic method:

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Critical Process for Finding the nth Term:

The key to success is following these steps in order - skipping steps or doing them out of sequence often leads to errors.

Step 1: Find the common difference

Look at the difference between consecutive terms. This should be the same throughout the sequence.

Step 2: Create the basic pattern

If the common difference is $d$ , then the nth term starts with $dn$ .

Step 3: Compare and adjust

Generate the sequence using $dn$ and compare it with the original sequence. Add or subtract a constant to match the original pattern.

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When comparing sequences, line up the terms vertically to make it easier to see what adjustment is needed. This visual comparison helps prevent calculation errors.

Worked examples

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Worked Example 1: Finding the first five terms

Given: $T_n = 4n - 3$

Solution:

$T_1 = 4(1) - 3 = 1$
$T_2 = 4(2) - 3 = 5$
$T_3 = 4(3) - 3 = 9$
$T_4 = 4(4) - 3 = 13$
$T_5 = 4(5) - 3 = 17$

Answer: The first 5 terms are: 1, 5, 9, 13, 17

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Worked Example 2: Finding the nth term from a sequence

Given sequence: 3, 7, 11, 15, ...

Solution:

Find the common difference: $7 - 3 = 4$ , $11 - 7 = 4$ , $15 - 11 = 4$
The difference is 4, so we start with $4n$
Generate the sequence $4n$ : 4, 8, 12, 16, ...
Compare with original:
- $4n$ gives us: 4, 8, 12, 16, ...
- Original is: 3, 7, 11, 15, ...
- We need to subtract 1 from each term: $4-1=3$ , $8-1=7$ , $12-1=11$ , $16-1=15$

Answer: $T_n = 4n - 1$

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Worked Example 3: Finding a specific term

Given sequence: 4, 7, 10, 13, 16, ... Find the 20th term.

Solution:

Common difference = 3
Start with $3n$ : this gives 3, 6, 9, 12, ...
Compare sequences:
- Original: 4, 7, 10, 13, ...
- $3n$ gives: 3, 6, 9, 12, ...
- Add 1 to each term: $3+1=4$ , $6+1=7$ , $9+1=10$ , $12+1=13$
Therefore: $T_n = 3n + 1$
Find $T_{20}$ : $T_{20} = 3(20) + 1 = 60 + 1 = 61$

Answer: The 20th term is 61

Key formulas for arithmetic sequences

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Essential Formulas to Remember:

For an arithmetic sequence with first term $a$ and common difference $d$ :

General form: $T_n = dn + (a - d)$
Alternative form: $T_n = a + (n - 1)d$

Both formulas will give the same result - choose the one that feels more natural to you!

Exam tips

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Exam Success Strategies:

Always check your nth term formula by substituting $n = 1, 2, 3$ to see if you get the correct first few terms
Remember that the coefficient of n in your formula should equal the common difference
Show all your working clearly, especially when comparing sequences
Double-check your arithmetic when calculating specific terms
If you're running short on time, at least show the method even if you don't complete all calculations

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Key Points to Remember:

The nth term is a formula that lets you find any term in a sequence without calculating all previous terms
For arithmetic sequences, find the common difference first, then use it to build your formula
Always compare sequences to find what constant to add or subtract
Check your answer by substituting back into your formula
The general form for arithmetic sequences is $T_n = dn ± \text{constant}$ , where $d$ is the common difference

The nth Term of a Sequence (Leaving Cert Mathematics): Revision Notes