Sequences & Patterns

<p>Sequences are lists of numbers that follow a rule. In the 11+ exam, you might be asked to find the next number, a missing number in the middle, or the rule itself. Getting good at spotting patterns quickly will save you valuable time.</p>

<h2>Arithmetic Sequences</h2>

<p>In an arithmetic sequence, the same number is added (or subtracted) each time. This number is called the <strong>common difference</strong>.</p>

<p><strong>Example:</strong> 5, 9, 13, 17, 21, ... The common difference is +4 (each number is 4 more than the last). The next number is 21 + 4 = 25.</p>

<p><strong>Example:</strong> 30, 25, 20, 15, ... The common difference is −5. The next number is 15 − 5 = 10.</p>

<h2>Finding the Rule</h2>

<p>Write down the differences between each pair of consecutive numbers. If the differences are all the same, you have an arithmetic sequence.</p>

<p><strong>Worked Example:</strong> 3, 7, 11, 15, ?, 23</p>

<p>Differences: 4, 4, 4, ?, ?. The rule is +4, so the missing number is 15 + 4 = 19. Check: 19 + 4 = 23. ✓</p>

<h2>Two-Step Rules</h2>

<p>Some sequences use a two-step rule such as "multiply by 2 then add 1."</p>

<p><strong>Example:</strong> 1, 3, 7, 15, 31, ... Each number is doubled and then 1 is added: (1 × 2 + 1 = 3), (3 × 2 + 1 = 7), (7 × 2 + 1 = 15). Next: 31 × 2 + 1 = 63.</p>

<h2>Geometric Sequences</h2>

<p>In a geometric sequence, each term is multiplied by the same number.</p>

<p><strong>Example:</strong> 2, 6, 18, 54, ... Each term is multiplied by 3. Next: 54 × 3 = 162.</p>

<h2>Square and Triangular Numbers</h2>

<p><strong>Square numbers:</strong> 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. These are 1², 2², 3², 4², and so on. Learn them up to 12² = 144.</p>

<p><strong>Triangular numbers:</strong> 1, 3, 6, 10, 15, 21, 28, 36, 45, 55. You get them by adding 1, then 2, then 3, then 4, and so on.</p>

<h2>Picture Patterns</h2>

<p>Sometimes a sequence is shown using shapes or dots. Count the objects in each pattern and write the numbers down as a sequence. Then find the rule as normal.</p>

<p><strong>Worked Example:</strong> Pattern 1 has 4 dots, Pattern 2 has 7 dots, Pattern 3 has 10 dots. How many dots in Pattern 6?</p>

<p>The sequence is 4, 7, 10, ... with a common difference of +3. Continue: 13, 16, 19. Pattern 6 has 19 dots.</p>

<h2>Finding the nth Term (Bonus Skill)</h2>

<p>For arithmetic sequences, the nth term rule is: <strong>first term + (n − 1) × common difference</strong>. For 4, 7, 10, ... the nth term is 4 + (n − 1) × 3 = 3n + 1. Pattern 6: 3(6) + 1 = 19. ✓</p>

<h2>Practice Tips</h2>

<ul>

<li>Always write out the differences between terms — do not try to spot the pattern just by looking</li>

<li>If the differences are not constant, look at the differences of the differences (second differences)</li>

<li>Memorise square numbers up to 144 and triangular numbers up to 55</li>

<li>Check your answer by applying the rule to the previous terms to see if they match</li>

</ul>