Finite Arithmetic Series Revision Notes for Grade 12 NSC Matric Mathematics

Finite Arithmetic Series

What is a finite arithmetic series?

An arithmetic sequence is a sequence of numbers where the difference between any term and the previous term is a constant number called the common difference (d). When we add up a finite number of terms from an arithmetic sequence, we get a finite arithmetic series.

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Understanding the difference between a sequence (the ordered list of numbers) and a series (the sum of those numbers) is essential for mastering this topic. A sequence lists the terms, while a series adds them together.

Key formula for arithmetic sequences

The nth term of an arithmetic sequence is given by:

$T_n = a + (n - 1)d$

Where:

$T_n$ is the nth term of the sequence
$a$ is the first term
$d$ is the common difference
$n$ is the position of the term

Gauss's method for summing arithmetic series

The mathematician Karl Friedrich Gauss discovered an ingenious method for adding arithmetic series when he was only 8 years old. His teacher asked the class to add all numbers from 1 to 100, expecting to keep them busy for a long time. Young Gauss quickly found the answer: 5050.

Gauss's approach

To find the sum of $1 + 2 + 3 + ... + 100$ :

Write the numbers in ascending order: $1 + 2 + 3 + ... + 98 + 99 + 100$
Write the numbers in descending order: $100 + 99 + 98 + ... + 3 + 2 + 1$
Add corresponding pairs together: Each pair sums to 101
Simplify: $2S_{100} = 101 × 100 = 10,100$ , so $S_{100} = 5050$

This method works because adding the first and last terms, second and second-last terms, etc., always gives the same result.

General formula for finite arithmetic series

Instead of adding term by term, we can use a general formula. Starting with an arithmetic sequence of n terms from the first term (a) to the last term (l):

$S_n = a + (a + d) + (a + 2d) + ... + (l - 2d) + (l - d) + l$

Adding the same series in reverse order: $S_n = l + (l - d) + (l - 2d) + ... + (a + 2d) + (a + d) + a$

Adding these two equations: $2S_n = (a + l) + (a + l) + (a + l) + ... + (a + l) + (a + l) + (a + l)$ $2S_n = n × (a + l)$

Therefore: $S_n = \frac{n}{2}(a + l)$

Alternative form

Since the last term $l = a + (n - 1)d$ , we can substitute this into our formula:

$S_n = \frac{n}{2}(a + [a + (n - 1)d])$ $S_n = \frac{n}{2}[2a + (n - 1)d]$

The two main formulas

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The Two Essential Formulas for Arithmetic Series:

$S_n = \frac{n}{2}[2a + (n - 1)d]$ (use when you know a, d, and n)

$S_n = \frac{n}{2}(a + l)$ (use when you know a, l, and n)

These formulas are particularly useful when dealing with large numbers of terms, as they save considerable time compared to adding individual terms.

Worked example 1: Using the general formula

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Worked Example: Finding the sum using the general formula

Question: Find the sum of the first 30 terms of an arithmetic series with $T_n = 7n - 5$ .

Solution:

Step 1: Find the sequence pattern and identify known variables

$T_1 = 7(1) - 5 = 2$
$T_2 = 7(2) - 5 = 9$
$T_3 = 7(3) - 5 = 16$

This gives the sequence: 2, 9, 16, ... So: $a = 2$ , $d = 7$ , $n = 30$

Step 2: Apply the general formula

$S_n = \frac{n}{2}[2a + (n - 1)d]$
$S_{30} = \frac{30}{2}[2(2) + (30 - 1)(7)]$
$S_{30} = 15(4 + 203) = 15(207) = 3105$

Answer: $S_{30} = 3105$

Worked example 2: When first and last terms are known

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Worked Example: Using first and last terms

Question: Find the sum of the series $-5 - 3 - 1 + ... + 123$ .

Solution:

Step 1: Identify the pattern and known variables

Check the common difference:

$d = T_2 - T_1 = -3 - (-5) = 2$
$d = T_3 - T_2 = -1 - (-3) = 2$

So: $a = -5$ , $d = 2$ , $l = 123$

Step 2: Find the number of terms

Using $T_n = a + (n - 1)d$ :

$123 = -5 + (n - 1)(2)$
$123 = -5 + 2n - 2$
$130 = 2n$
$n = 65$

Step 3: Use the appropriate formula

Since we know a, l, and n, use: $S_n = \frac{n}{2}(a + l)$

$S_{65} = \frac{65}{2}(-5 + 123) = \frac{65}{2}(118) = 3835$

Answer: $S_{65} = 3835$

Worked example 3: Finding the number of terms

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Worked Example: Determining the number of terms

Question: Given an arithmetic sequence with $T_2 = 7$ and $d = 3$ , determine how many terms must be added together to give a sum of 2146.

Solution:

Step 1: Find the first term

Since $d = T_2 - T_1$ :

$3 = 7 - a$
$a = 4$

Known values: $a = 4$ , $d = 3$ , $S_n = 2146$

Step 2: Use the general formula to find n

$S_n = \frac{n}{2}[2a + (n - 1)d]$
$2146 = \frac{n}{2}[2(4) + (n - 1)(3)]$
$4292 = n(8 + 3n - 3)$
$4292 = n(5 + 3n)$
$4292 = 5n + 3n^2$
$3n^2 + 5n - 4292 = 0$

Factorising: $(3n + 116)(n - 37) = 0$

So $n = -\frac{116}{3}$ or $n = 37$

Since n must be positive: $n = 37$

Finding individual terms from sums

Sometimes you need to find a specific term when you know partial sums of a series. This is a common exam scenario that requires understanding the relationship between consecutive sums.

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Key Relationship for Finding Individual Terms:

For any arithmetic series: $T_n = S_n - S_{n-1}$ (for $n ≥ 2$ ) And: $T_1 = S_1$

This works because:

$S_2 = T_1 + T_2$ , so $T_2 = S_2 - T_1 = S_2 - S_1$
$S_3 = T_1 + T_2 + T_3$ , so $T_3 = S_3 - S_2$
And so on...

Practical application

If you know $S_1 = 5$ , $S_2 = 12$ , $S_3 = 21$ , then:

$T_1 = S_1 = 5$
$T_2 = S_2 - S_1 = 12 - 5 = 7$
$T_3 = S_3 - S_2 = 21 - 12 = 9$

Exam tips and common pitfalls

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Critical Points to Avoid Common Mistakes:

When to use each formula:

Use $S_n = \frac{n}{2}[2a + (n - 1)d]$ when you know the first term, common difference, and number of terms
Use $S_n = \frac{n}{2}(a + l)$ when you know the first term, last term, and number of terms

Common mistakes to avoid:

Always check that the sequence is actually arithmetic by verifying the common difference is constant
When solving quadratic equations for n, remember that n must be a positive integer
Double-check your arithmetic, especially with negative numbers
Make sure you identify what the question is asking for (sum, number of terms, specific term, etc.)

Problem-solving strategy:

Identify what type of problem you're dealing with
Write down all known information clearly
Choose the appropriate formula based on what you know
Substitute carefully and solve step by step
Check that your answer makes sense in context

Remember!

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Key Points to Remember:

An arithmetic sequence has a constant difference between consecutive terms
A finite arithmetic series is the sum of a finite number of terms from an arithmetic sequence
The two main formulas are: $S_n = \frac{n}{2}[2a + (n - 1)d]$ and $S_n = \frac{n}{2}(a + l)$
Gauss's method pairs terms from opposite ends to find patterns in sums
Individual terms can be found using $T_n = S_n - S_{n-1}$ when partial sums are known

Finite Arithmetic Series (Grade 12 NSC Matric Mathematics): Revision Notes