Fix the mistake: Patterns

Patterns ① Sequences That Increase by the Same Amount and Sequences That Multiply by the Same Factor

A row of numbers is called a “sequence.” Many sequences follow a rule in how their numbers are arranged.
A sequence in which the “difference” between neighboring terms is always the same is called an arithmetic sequence, and a sequence in which the “multiplication factor (ratio)” between neighboring terms is always the same is called a geometric sequence.

What You Will Learn

Example of an arithmetic sequence: 2, 5, 8, 11, 14, … (the difference between neighboring terms is always 3)
　nth term ＝ first term ＋ (n－1) × difference (common difference)

Example of a geometric sequence: 3, 6, 12, 24, 48, … (each term is always twice the previous term)
　nth term ＝ first term × (common ratio) multiplied (n－1) times

Example Problem
For the arithmetic sequence 3, 7, 11, 15, …, find the 20th term and the sum of the 1st through 20th terms.

Solution:
① Looking at the difference between neighboring terms, the sequence increases by 4 each time (difference = 4).
② Calculate the 20th term.
　3 ＋ (20－1) × 4 ＝ 3 ＋ 76 ＝ 79
③ The sum of the 1st through 20th terms can be found using “(first term ＋ last term) × number of terms ÷ 2.”
　(3 ＋ 79) × 20 ÷ 2 ＝ 82 × 10 ＝ 820

Key Points

Remember
  • Sum of an arithmetic sequence ＝ (first term ＋ last term) × number of terms ÷ 2
  • For a geometric sequence, first check “what factor the terms are multiplied by each time.”
  • When counting “which term,” count the first term as the 1st term.

Practice Problems

  1. Find the 15th term of the arithmetic sequence 5, 9, 13, 17, … .
  2. Find the sum of the 1st through 10th terms of the arithmetic sequence 4, 7, 10, … .
  3. Find the 6th term of the geometric sequence 2, 6, 18, 54, … .

Answers and Explanations

  1. Answer: 15th term = 61
    Explanation: This is an arithmetic sequence with first term 5 and common difference 4. The nth term = 5 + (n-1)×4 = 4n+1. The 15th term = 4×15+1 = 61.
  2. Answer: Sum from the 1st term through the 10th term = 175
    Explanation: This is an arithmetic sequence with first term 4 and common difference 3. The 10th term = 4+(10-1)×3 = 4+27 = 31. Sum = (first term+last term)×number of terms÷2 = (4+31)×10÷2 = 35×5 = 175.
  3. Answer: 6th term = 486
    Explanation: This is a geometric sequence with first term 2 and common ratio 3. The nth term = 2×3^(n-1). The 6th term = 2×3^5 = 2×243 = 486.

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