Geometric Sequences and Pattern Identification

Sequence Analysis and Pattern Identification

• Definition of the Sequence:

  • The given numerical sequence consists of the terms: $2, 4, 8, 16, \dots$
  • Individual terms in the sequence are defined as:
  • First term ($a_1$): $2$
  • Second term ($a_2$): $4$
  • Third term ($a_3$): $8$
  • Fourth term ($a_4$): $16$

• Identifying the Pattern:

  • Geometric Multiplicative Pattern:

  • Each consecutive term is generated by multiplying the preceding term by a constant multiplier (common ratio, $r = 2$):       $a_2 = 2 \times 2 = 4$$a_3 = 4 \times 2 = 8$$a_4 = 8 \times 2 = 16$

  • The general term formula for the $n$-th term of this sequence is expressed as an exponential power of 2:       $a_n = 2^n$

  • Additive Difference Pattern:

  • Examining the differences between consecutive terms reveals:       $a_2 - a_1 = 4 - 2 = +2$$a_3 - a_2 = 8 - 4 = +4$$a_4 - a_3 = 16 - 8 = +8$

  • The sequence of differences is $2, 4, 8, \dots$, which doubles at each step.

Step-by-Step Solution and Evaluation

• Problem Statement:

  • Task: Write down the next number in the sequence $2, 4, 8, 16, \dots$
  • Allocated Marks: $1\text{ mark}$

• Calculating the Next Term ($a_5$):

  • Method 1: Geometric Multiplication:
  • Multiply the fourth term ($16$) by the common ratio of $2$:       $a_5 = 16 \times 2 = 32$
  • Method 2: Exponential Power:
  • Substitute $n = 5$ into the $n$-th term formula $a_n = 2^n$:       $a_5 = 2^5 = 32$
  • Method 3: Pattern of Differences:
  • The next difference must be double the previous difference ($+8 \times 2 = +16$):       $a_5 = 16 + 16 = 32$

• Correct Final Answer:

  • The next number in the sequence is $32$

Examination Sheet Details and Error Analysis

• Document Details:

  • Previous question completed answer: $13$
  • Target question text: "Write down the next number in the sequence"
  • Provided sequence: $2, 4, 8, 16$
  • Annotations added under sequence:
  • Between $2$ and $4$: $+2$
  • Between $4$ and $8$: $+4$
  • Between $8$ and $16$: $+8$
  • Recorded student answer: $20$

• Misconception and Error Analysis:

  • Recorded Answer: $20$
  • Error Breakdown:
  • The annotated steps accurately recognize the first three term-to-term additions ($+2$, $+4$, $+8$).
  • To arrive at $20$, the student added $4$ to the fourth term ($16 + 4 = 20$) instead of continuing the doubling pattern ($+16$ or $\times 2$).
  • This error represents a breakdown in identifying non-linear sequence progression, confusing geometric doubling increments with constant linear addition.