Tile Pattern Analysis and Arithmetic Sequence Calculation

Table Data and Initial Observations

• Figure $1$ contains $5$ tiles.

• Figure $2$ contains $9$ tiles.

• Figure $3$ contains $13$ tiles.

• Figure $4$ contains $17$ tiles.

• The sequence of tile counts corresponding to the sequential figure numbers is $5$, $9$, $13$, $17$, $\dots$

Identification of Sequence Type and Common Difference

• The difference between consecutive terms is evaluated as follows:

  • Difference between Figure $2$ and Figure $1$: $9 - 5 = 4$

  • Difference between Figure $3$ and Figure $2$: $13 - 9 = 4$

  • Difference between Figure $4$ and Figure $3$: $17 - 13 = 4$

• Because the difference between consecutive figure outputs remains constant ($d = 4$), the system represents an arithmetic sequence (a linear pattern).

• The initial value or first term is $a_1 = 5$.

• The common difference between consecutive figures is $d = 4$

Explicit Formula Derivation

• The general formula for the $n$\text{-th} term of an arithmetic sequence is given by: $a_n = a_1 + (n - 1) \times d$

• Substituting the established parameters ($a_1 = 5$ and $d = 4$) into the standard formula gives: $a_n = 5 + (n - 1) \times 4$

• Distributing the common difference $4$ across $(n - 1)$ yields: $a_n = 5 + 4 \times n - 4$

• Combining like terms simplifies the relation to the explicit linear equation: $a_n = 4 \times n + 1$

Verification of the Explicit Formula

• Verification for Figure $1$ ($n = 1$): $a_1 = 4 \times 1 + 1 = 5$

• Verification for Figure $2$ ($n = 2$): $a_2 = 4 \times 2 + 1 = 9$

• Verification for Figure $3$ ($n = 3$): $a_3 = 4 \times 3 + 1 = 13$

• Verification for Figure $4$ ($n = 4$): $a_4 = 4 \times 4 + 1 = 17$

Calculation for Figure 10

• Method 1: Explicit Formula Evaluation

  • To find the number of tiles in Figure $10$, substitute $n = 10$ into the derived formula $a_n = 4 \times n + 1$:   $a_{10} = 4 \times 10 + 1$   $a_{10} = 40 + 1$   $a_{10} = 41$

• Method 2: Sequential Recursive Extension

  • Figure $1$: $5$\,\text{tiles}

  • Figure $2$: $5 + 4 = 9$\,\text{tiles}

  • Figure $3$: $9 + 4 = 13$\,\text{tiles}

  • Figure $4$: $13 + 4 = 17$\,\text{tiles}

  • Figure $5$: $17 + 4 = 21$\,\text{tiles}

  • Figure $6$: $21 + 4 = 25$\,\text{tiles}

  • Figure $7$: $25 + 4 = 29$\,\text{tiles}

  • Figure $8$: $29 + 4 = 33$\,\text{tiles}

    

  • Figure $9$: $33 + 4 = 37$\,\text{tiles}

  • Figure $10$: $37 + 4 = 41$\,\text{tiles}

• Total tile count for Figure $10$ is $41$\,\text{tiles}.