Grade 11

Advanced

7 min

Lesson 7 of 12

Arithmetic Sequences and Series

Not Started

Study patterns with constant differences and learn to sum large sets of linear data.

Imagine you are building a pyramid where each row has exactly two more blocks than the one above it. If the top row has 1 block and there are 100 rows, how many blocks do you need in total? Arithmetic series allow you to solve this in seconds without counting a single brick.

Learning Objectives

1.Derive and apply the formula for the $n$ th term of an arithmetic sequence
2.Calculate the partial sum of an arithmetic series using sigma notation
3.Solve complex real-world accumulation problems using linear growth patterns

The DNA of Arithmetic Sequences

An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant is called the common difference (

d

). If you know the first term (

a_1

) and the difference (

d

), you can find any term in the universe using the formula:

a_n = a_1 + (n-1)d

. Here,

n

represents the position of the term. We subtract 1 from

n

because we don't apply the difference to the first term; we start adding

d

only from the second term onwards. This linear relationship is the foundation of all predictable growth patterns.

Find the 50th term of the sequence: $7, 12, 17, 22, ...$

1. Identify $a_1$ : The first term is $7$ . 2. Identify $d$ : $12 - 7 = 5$ . 3. Use the formula: $a_{50} = 7 + (50-1)5$ . 4. Calculate: $a_{50} = 7 + (49 \times 5) = 7 + 245 = 252$ .

Quick Check

In the sequence $10, 6, 2, -2, ...$ , what is the common difference $d$ ?

Subtract the first term from the second term.

Answer

$d = -4$

The Power of Summation

When we add the terms of an arithmetic sequence together, we create an arithmetic series. Legend has it that a young Carl Friedrich Gauss discovered the formula for this when his teacher asked the class to sum the numbers 1 to 100. He realized that pairing the first and last terms (

1+100, 2+99, ...

) always yielded the same sum (

101

). The formula for the partial sum (

S_n

) of the first

n

terms is:

S_n = \frac{n}{2}(a_1 + a_n)

or, by substituting the

a_n

formula:

S_n = \frac{n}{2}[2a_1 + (n-1)d]

A consultant earns $\$ 50,000 $in year 1, and receives a$ \ $3,000$ raise every year. How much total income will they have earned after 10 years?

1. Identify variables: $a_1 = 50000$ , $d = 3000$ , $n = 10$ . 2. Choose formula: $S_n = \frac{n}{2}[2a_1 + (n-1)d]$ . 3. Substitute: $S_{10} = \frac{10}{2}[2(50000) + (9)(3000)]$ . 4. Solve: $S_{10} = 5[100000 + 27000] = 5[127000] = \$ 635,000$.

Quick Check

If the first term is 5 and the 10th term is 45, what is the sum of these 10 terms?

Use

S_n = \frac{n}{2}(a_1 + a_n)

Answer

250

Decoding Sigma Notation

In advanced mathematics, we use Sigma Notation ( $\sum$ ) to write series concisely. The symbol $\sum_{k=1}^{n} f(k)$ tells you to start at $k=1$ , plug every integer up to $n$ into the function $f(k)$ , and add them all up. For an arithmetic series, the function $f(k)$ will always be linear (e.g., $3k + 2$ ). To solve these, simply find the first term (plug in the bottom number), the last term (plug in the top number), and the number of terms ( $n = \text{top} - \text{bottom} + 1$ ).

Evaluate: $\sum_{k=5}^{20} (4k - 2)$

1. Find $n$ : $20 - 5 + 1 = 16$ terms. 2. Find $a_1$ (using $k=5$ ): $4(5) - 2 = 18$ . 3. Find $a_n$ (using $k=20$ ): $4(20) - 2 = 78$ . 4. Use the sum formula: $S_{16} = \frac{16}{2}(18 + 78) = 8(96) = 768$ .

Key Takeaways

1An arithmetic sequence changes by a constant common difference ( $d$ ).
2The $n$ th term formula is $a_n = a_1 + (n-1)d$ .
3The sum of an arithmetic series is the average of the first and last terms, multiplied by the number of terms: $S_n = \frac{n}{2}(a_1 + a_n)$ .
4Sigma notation is a shorthand for summation where the expression inside is a linear function of the index.

Quick Check

Multiple Choice

What is the 11th term of the sequence where $a_1 = -5$ and $d = 3$ ?

Multiple Choice

Evaluate $\sum_{i=1}^{100} i$ .

True/False

The common difference $d$ must always be a positive integer.

What's Next?

Review Tomorrow

In 24 hours, try to write down the two versions of the arithmetic sum formula from memory and explain why the $(n-1)$ exists in the $n$ th term formula.

Practice Activity

Find the sum of all multiples of 7 between 100 and 500. Hint: Find the first multiple ( $a_1$ ) and the last multiple ( $a_n$ ) first!

Browse

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Arithmetic Sequences and Series

Learning Objectives

The DNA of Arithmetic Sequences

The Power of Summation

Decoding Sigma Notation

Key Takeaways

Quick Check

What's Next?

Arithmetic Sequences and Series

Learning Objectives

The DNA of Arithmetic Sequences

The Power of Summation

Decoding Sigma Notation

Key Takeaways

Quick Check

What's Next?