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Amy Ferguson Moncure

IB AI · Unit 2 · 2A Arithmetic Sequences

Find Formula

Substitute the first term and the common difference, then simplify until uₙ is written in terms of n.

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Key Ideas Given u₁ and d First Three Terms In Words

SL 1.2

\(u_n = u_1 + (n-1)d\)

\(S_n = \dfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr)\)

\(S_n = \dfrac{n}{2}(u_1 + u_n)\)

What to Remember

The nth-term formula

\(u_n = u_1 + (n-1)d\)

\(u_n\) = the formula

\(u_1\) = first term

\(n\) = the term number

\(d\) = common difference

Example. \(u_1 = 5\), \(d = 4\). Write a formula for \(u_n\).

\(u_1 = 5\)

\(d = 4\)

\(u_n = 5 + (n-1) \times 4\)

\(u_n = 5 + 4n - 4\)

\(u_n = 4n + 1\)

Easy 1

Given u₁ and d

Substitute, then simplify.

Example
Write a formula for the nth term.\(u_1 = 8\)\(d = 3\)Write a formula for \(u_n\).

\(u_n = u_1 + (n-1)d\)

\(u_1 = 8\)

\(d = 3\)

\(u_n = 8 + (n-1) \times 3\)

\(u_n = 8 + 3n - 3\)

\(u_n = 3n + 5\)

Answer: \(u_n = 3n + 5\)

1
Write a formula for the nth term.\(u_1 = 7\)\(d = 3\)Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 7\)

\(d = 3\)

\(u_n = 7 + (n-1) \times 3\)

\(u_n = 7 + 3n - 3\)

\(u_n = 3n + 4\)

Answer: \(u_n = 3n + 4\)

2
Write a formula for the nth term.\(u_1 = 2\)\(d = 6\)Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 2\)

\(d = 6\)

\(u_n = 2 + (n-1) \times 6\)

\(u_n = 2 + 6n - 6\)

\(u_n = 6n - 4\)

Answer: \(u_n = 6n - 4\)

3
Write a formula for the nth term.\(u_1 = 10\)\(d = 2\)Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 10\)

\(d = 2\)

\(u_n = 10 + (n-1) \times 2\)

\(u_n = 10 + 2n - 2\)

\(u_n = 2n + 8\)

Answer: \(u_n = 2n + 8\)

4
Write a formula for the nth term.\(u_1 = 15\)\(d = -3\)Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 15\)

\(d = -3\)

\(u_n = 15 + (n-1) \times \left(-3\right)\)

\(u_n = 15 - 3(n-1)\)

\(u_n = 15 - 3n + 3\)

\(u_n = 18 - 3n\)

Answer: \(u_n = 18 - 3n\)

5
Write a formula for the nth term.\(u_1 = 4\)\(d = 5\)Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 4\)

\(d = 5\)

\(u_n = 4 + (n-1) \times 5\)

\(u_n = 4 + 5n - 5\)

\(u_n = 5n - 1\)

Answer: \(u_n = 5n - 1\)

Easy 2

First Three Terms

Find the first term and the common difference, then the formula.

Example
The first three terms of an arithmetic sequence are 6, 10, and 14. Write a formula for \(u_n\).

The first term is the first number. The common difference is the jump from one term to the next.

\(u_1 = 6\)

\(d = 10 - 6\)

\(d = 4\)

\(u_n = u_1 + (n-1)d\)

\(u_1 = 6\)

\(d = 4\)

\(u_n = 6 + (n-1) \times 4\)

\(u_n = 6 + 4n - 4\)

\(u_n = 4n + 2\)

Answer: \(u_n = 4n + 2\)

1
The first three terms of an arithmetic sequence are 3, 8, and 13. Write a formula for \(u_n\).
Show Answer

The first term is the first number. The common difference is the jump from one term to the next.

\(u_1 = 3\)

\(d = 8 - 3\)

\(d = 5\)

\(u_n = u_1 + (n-1)d\)

\(u_1 = 3\)

\(d = 5\)

\(u_n = 3 + (n-1) \times 5\)

\(u_n = 3 + 5n - 5\)

\(u_n = 5n - 2\)

Answer: \(u_n = 5n - 2\)

2
The first three terms of an arithmetic sequence are 9, 12, and 15. Write a formula for \(u_n\).
Show Answer

The first term is the first number. The common difference is the jump from one term to the next.

\(u_1 = 9\)

\(d = 12 - 9\)

\(d = 3\)

\(u_n = u_1 + (n-1)d\)

\(u_1 = 9\)

\(d = 3\)

\(u_n = 9 + (n-1) \times 3\)

\(u_n = 9 + 3n - 3\)

\(u_n = 3n + 6\)

Answer: \(u_n = 3n + 6\)

3
The first three terms of an arithmetic sequence are 20, 16, and 12. Write a formula for \(u_n\).
Show Answer

The first term is the first number. The common difference is the jump from one term to the next.

\(u_1 = 20\)

\(d = 16 - 20\)

\(d = -4\)

\(u_n = u_1 + (n-1)d\)

\(u_1 = 20\)

\(d = -4\)

\(u_n = 20 + (n-1) \times \left(-4\right)\)

\(u_n = 20 - 4(n-1)\)

\(u_n = 20 - 4n + 4\)

\(u_n = 24 - 4n\)

Answer: \(u_n = 24 - 4n\)

4
The first three terms of an arithmetic sequence are 1, 7, and 13. Write a formula for \(u_n\).
Show Answer

The first term is the first number. The common difference is the jump from one term to the next.

\(u_1 = 1\)

\(d = 7 - 1\)

\(d = 6\)

\(u_n = u_1 + (n-1)d\)

\(u_1 = 1\)

\(d = 6\)

\(u_n = 1 + (n-1) \times 6\)

\(u_n = 1 + 6n - 6\)

\(u_n = 6n - 5\)

Answer: \(u_n = 6n - 5\)

5
The first three terms of an arithmetic sequence are 8, 11, and 14. Write a formula for \(u_n\).
Show Answer

The first term is the first number. The common difference is the jump from one term to the next.

\(u_1 = 8\)

\(d = 11 - 8\)

\(d = 3\)

\(u_n = u_1 + (n-1)d\)

\(u_1 = 8\)

\(d = 3\)

\(u_n = 8 + (n-1) \times 3\)

\(u_n = 8 + 3n - 3\)

\(u_n = 3n + 5\)

Answer: \(u_n = 3n + 5\)

Medium 1

In Words

Read off the first term and the common difference, then write the formula.

Example
An arithmetic sequence increases by the same amount each term. The first term is 11 and the common difference is 4. Write a formula for \(u_n\).

\(u_n = u_1 + (n-1)d\)

\(u_1 = 11\)

\(d = 4\)

\(u_n = 11 + (n-1) \times 4\)

\(u_n = 11 + 4n - 4\)

\(u_n = 4n + 7\)

Answer: \(u_n = 4n + 7\)

1
An arithmetic sequence increases by the same amount each term. The first term is 6 and the common difference is 5. Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 6\)

\(d = 5\)

\(u_n = 6 + (n-1) \times 5\)

\(u_n = 6 + 5n - 5\)

\(u_n = 5n + 1\)

Answer: \(u_n = 5n + 1\)

2
An arithmetic sequence decreases by the same amount each term. The first term is 30 and the common difference is -4. Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 30\)

\(d = -4\)

\(u_n = 30 + (n-1) \times \left(-4\right)\)

\(u_n = 30 - 4(n-1)\)

\(u_n = 30 - 4n + 4\)

\(u_n = 34 - 4n\)

Answer: \(u_n = 34 - 4n\)

3
An arithmetic sequence increases by the same amount each term. The first term is 2 and the common difference is 9. Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 2\)

\(d = 9\)

\(u_n = 2 + (n-1) \times 9\)

\(u_n = 2 + 9n - 9\)

\(u_n = 9n - 7\)

Answer: \(u_n = 9n - 7\)

4
An arithmetic sequence decreases by the same amount each term. The first term is 18 and the common difference is -2. Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 18\)

\(d = -2\)

\(u_n = 18 + (n-1) \times \left(-2\right)\)

\(u_n = 18 - 2(n-1)\)

\(u_n = 18 - 2n + 2\)

\(u_n = 20 - 2n\)

Answer: \(u_n = 20 - 2n\)

5
An arithmetic sequence increases by the same amount each term. The first term is 7 and the common difference is 8. Write a formula for \(u_n\).
Show Answer

\(u_n = u_1 + (n-1)d\)

\(u_1 = 7\)

\(d = 8\)

\(u_n = 7 + (n-1) \times 8\)

\(u_n = 7 + 8n - 8\)

\(u_n = 8n - 1\)

Answer: \(u_n = 8n - 1\)

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