Sequences | Free ACT Math Course | The School of Mathematics

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<h1>ACT Math: Sequences</h1>

Sequences is a Functions-domain topic on the ACT, testing your ability to spot a pattern and express it as a formula. This free, complete lesson covers arithmetic sequences, geometric sequences, recursive formulas, and the sum of finite arithmetic and geometric series. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-sequences-quiz-1">Practice Sequences Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#arithmetic">Arithmetic Sequences</a></li>

<li><a href="#geometric">Geometric Sequences</a></li>

<li><a href="#recursive">Recursive Formulas</a></li>

<li><a href="#sums">Sum of a Finite Series</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

</nav>

<h2 id="arithmetic">1. Arithmetic Sequences</h2>

<p>An <strong>arithmetic sequence</strong> has a constant difference, called the <strong>common difference</strong> (d), between consecutive terms. Its explicit (nth-term) formula is:</p>

<p><strong>Worked Example:</strong> Find the 15th term of the arithmetic sequence 4, 9, 14, 19, ...</p>

<p>The common difference is d = 5.</p>

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<p class="prompt">1. Find the 10th term of the arithmetic sequence 3, 7, 11, 15, ...</p>

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<p><span class="label">Explanation:</span> d = 4. a<sub>10</sub> = 3 + (10 − 1)(4) = 3 + 36 = 39.</p>

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<p class="prompt">2. Find the common difference of the sequence 20, 14, 8, 2, ...</p>

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<p><span class="label">Explanation:</span> d = 14 − 20 = −6 (each term decreases by 6).</p>

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<p class="prompt">3. Write an explicit formula for the sequence 5, 8, 11, 14, ...</p>

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<p><span class="label">Explanation:</span> a<sub>1</sub> = 5, d = 3. a<sub>n</sub> = 5 + (n − 1)(3) = 5 + 3n − 3 = 3n + 2.</p>

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<p class="prompt">4. In an arithmetic sequence, the 3rd term is 17 and the 7th term is 37. What is the common difference?</p>

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<p><span class="label">Explanation:</span> Going from the 3rd term to the 7th term takes 4 steps and changes the value by 37 − 17 = 20. d = 20 ÷ 4 = 5.</p>

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<p class="prompt">5. Using the sequence from the previous problem (3rd term = 17, common difference = 5), what is the 1st term?</p>

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<p><span class="label">Explanation:</span> a<sub>3</sub> = a<sub>1</sub> + 2d, so 17 = a<sub>1</sub> + 2(5), giving a<sub>1</sub> = 17 − 10 = 7.</p>

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<h2 id="geometric">2. Geometric Sequences</h2>

<p>A <strong>geometric sequence</strong> has a constant ratio, called the <strong>common ratio</strong> (r), between consecutive terms. Its explicit formula is:</p>

<p><strong>Worked Example:</strong> Find the 6th term of the geometric sequence 3, 6, 12, 24, ...</p>

<p>The common ratio is r = 2.</p>

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<p class="prompt">1. Find the common ratio of the sequence 5, 15, 45, 135, ...</p>

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<p><span class="label">Explanation:</span> r = 15 ÷ 5 = 3.</p>

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<p class="prompt">2. Find the 5th term of the geometric sequence 2, 6, 18, 54, ...</p>

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<p><span class="label">Explanation:</span> r = 3. a<sub>5</sub> = 2 × 3⁴ = 2 × 81 = 162.</p>

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<p class="prompt">3. Write an explicit formula for the sequence 4, 8, 16, 32, ...</p>

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<p><span class="label">Explanation:</span> a<sub>1</sub> = 4 and r = 2, giving a<sub>n</sub> = 4 × 2<sup>n − 1</sup>.</p>

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<p class="prompt">4. A geometric sequence has first term 5 and common ratio <span class="frac"><span class="num">1</span><span class="den">2</span></span>. What is the 4th term?</p>

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<p><span class="label">Explanation:</span> a<sub>4</sub> = 5 × <span class="frac"><span class="num">1</span><span class="den">2</span></span><sup>3</sup> = 5 × <span class="frac"><span class="num">1</span><span class="den">8</span></span> = <span class="frac"><span class="num">5</span><span class="den">8</span></span>.</p>

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<p class="prompt">5. In a geometric sequence, the 2nd term is 6 and the 3rd term is 18. What is the common ratio?</p>

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<p><span class="label">Explanation:</span> r = 18 ÷ 6 = 3.</p>

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<h2 id="recursive">3. Recursive Formulas</h2>

<p>A <strong>recursive formula</strong> defines each term using the previous term, rather than a direct formula in terms of n. Arithmetic sequences follow a<sub>n</sub> = a<sub>n−1</sub> + d; geometric sequences follow a<sub>n</sub> = r × a<sub>n−1</sub>.</p>

<p><strong>Worked Example:</strong> A sequence is defined recursively by a<sub>1</sub> = 7, a<sub>n</sub> = a<sub>n−1</sub> + 4 for n ≥ 2. Find the first 4 terms.</p>

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<p class="prompt">1. A sequence is defined by a<sub>1</sub> = 10, a<sub>n</sub> = a<sub>n−1</sub> − 3. What is a<sub>4</sub>?</p>

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<p><span class="label">Explanation:</span> a<sub>1</sub> = 10, a<sub>2</sub> = 7, a<sub>3</sub> = 4, a<sub>4</sub> = 1.</p>

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<p class="prompt">2. A sequence is defined by a<sub>1</sub> = 2, a<sub>n</sub> = 3 × a<sub>n−1</sub>. What is a<sub>4</sub>?</p>

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<p><span class="label">Explanation:</span> a<sub>1</sub> = 2, a<sub>2</sub> = 6, a<sub>3</sub> = 18, a<sub>4</sub> = 54.</p>

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<p class="prompt">3. The recursive formula a<sub>n</sub> = a<sub>n−1</sub> + d describes what type of sequence?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-3',true)">A) Arithmetic</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">B) Geometric</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">C) Neither</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">D) Both arithmetic and geometric</button>

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<p><span class="label">Explanation:</span> Adding a constant value (d) to get the next term is the defining feature of an arithmetic sequence.</p>

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<p class="prompt">4. The recursive formula a<sub>n</sub> = r × a<sub>n−1</sub> describes what type of sequence?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-4',true)">A) Geometric</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">B) Arithmetic</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">C) Neither</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">D) Both arithmetic and geometric</button>

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<p><span class="label">Explanation:</span> Multiplying by a constant ratio (r) to get the next term is the defining feature of a geometric sequence.</p>

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<h2 id="sums">4. Sum of a Finite Series</h2>

<p>The sum of the first n terms of an <strong>arithmetic</strong> sequence:</p>

<p>For a short geometric sequence, it's often just as fast to add the terms directly.</p>

<p><strong>Worked Example:</strong> Find the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15, ...</p>

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<p class="prompt">1. Find the sum of the first 20 terms of the arithmetic sequence 5, 9, 13, 17, ...</p>

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<p><span class="label">Explanation:</span> a<sub>20</sub> = 5 + 19(4) = 81. S<sub>20</sub> = <span class="frac"><span class="num">20</span><span class="den">2</span></span>(5 + 81) = 10(86) = 860.</p>

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<p class="prompt">2. Find the sum of the first 15 terms of an arithmetic sequence with a<sub>1</sub> = 2 and d = 3.</p>

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<p><span class="label">Explanation:</span> a<sub>15</sub> = 2 + 14(3) = 44. S<sub>15</sub> = <span class="frac"><span class="num">15</span><span class="den">2</span></span>(2 + 44) = 7.5(46) = 345.</p>

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<p class="prompt">3. What is the sum of the integers from 1 to 100?</p>

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<p><span class="label">Explanation:</span> This is an arithmetic sequence with a<sub>1</sub> = 1, a<sub>100</sub> = 100, and n = 100: S = <span class="frac"><span class="num">100</span><span class="den">2</span></span>(1 + 100) = 50(101) = 5,050.</p>

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<p class="prompt">4. Find the sum of the first 4 terms of the geometric sequence 3, 6, 12, 24.</p>

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<p><span class="label">Explanation:</span> 3 + 6 + 12 + 24 = 45.</p>

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<p class="prompt">5. Find the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, 162.</p>

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<p><span class="label">Explanation:</span> 2 + 6 + 18 + 54 + 162 = 242.</p>

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<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Using n instead of (n − 1) in the explicit formula.</strong> Both the arithmetic and geometric formulas use (n − 1), since the first term itself doesn't require any steps of change applied to it yet.</li>

<li><strong>Confusing an arithmetic and a geometric sequence.</strong> Check whether consecutive terms share a constant difference (arithmetic) or a constant ratio (geometric), the two patterns look very different once you check the actual numbers.</li>

<li><strong>Assuming every sequence with growing terms is geometric.</strong> An arithmetic sequence with a large common difference can also grow quickly; the defining test is always constant difference versus constant ratio, not how fast the numbers increase.</li>

<li><strong>Mixing up a recursive formula with an explicit formula.</strong> A recursive formula needs the previous term to compute the next one; an explicit formula computes any term directly from n, without needing prior terms.</li>

<li><strong>Forgetting that the arithmetic sum formula needs the last term of that specific sum, not the sequence's general nth term formula.</strong> Find a<sub>n</sub> for the exact number of terms you're summing before plugging into the sum formula.</li>

</ul>

<h2 id="faq">Frequently Asked Questions</h2>

<h3>How can I quickly tell if a sequence is arithmetic, geometric, or neither?</h3>

<p>Subtract consecutive terms; if the difference is always the same, it's arithmetic. If that fails, divide consecutive terms; if the ratio is always the same, it's geometric. If neither pattern holds, the sequence follows some other rule.</p>

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<h3>Do I need to memorize the sum formulas, or can I just add the terms directly?</h3>

<p>For a small number of terms, adding directly is often just as fast, especially for geometric sequences. For larger arithmetic sequences (dozens or hundreds of terms), the formula S<sub>n</sub> = <span class="frac"><span class="num">n</span><span class="den">2</span></span>(a<sub>1</sub> + a<sub>n</sub>) is far quicker and worth knowing.</p>

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<h3>What real-world situations use geometric sequences?</h3>

<p>Anything involving repeated percentage growth or decay follows a geometric pattern: compound interest, population growth at a constant rate, or a substance's radioactive decay, since each step multiplies the previous amount by the same factor.</p>

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<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/act-sequences-quiz-1">Sequences quizzes</a> in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-sequences-quiz-1">Practice Sequences Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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