1. **Stating the problem:** We have two sequences given: - Sequence A: $-1.23, 1.23, -1.23, 1.23, -1.23, 1.23, \ldots$ - Sequence B: $\frac{1}{4}, -\frac{1}{4}, \frac{1}{4}, -\frac{1}{4}, \frac{1}{4}, -\frac{1}{4}, \ldots$ We also have a bar graph representing terms 1 through 5 with lengths proportional to $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$ respectively. We want to understand how to analyze these sequences and the pattern shown in the bar graph. 2. **Understanding the sequences:** - Sequence A alternates between $-1.23$ and $1.23$. - Sequence B alternates between $\frac{1}{4}$ and $-\frac{1}{4}$. Both sequences are examples of alternating sequences where the sign changes every term. 3. **Formula for alternating sequences:** An alternating sequence can be written as: $$ a_n = (-1)^{n+1} \times c $$ where $c$ is the absolute value of the term. For Sequence A: $$ a_n = (-1)^{n+1} \times 1.23 $$ For Sequence B: $$ b_n = (-1)^{n+1} \times \frac{1}{4} $$ 4. **Understanding the bar graph:** The bar graph shows terms with lengths proportional to $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$. This suggests the terms are: $$ t_n = \frac{1}{n} $$ for $n=1,2,3,4,5$. 5. **Combining alternating sign and bar graph pattern:** If the terms alternate in sign and decrease in magnitude as $\frac{1}{n}$, the general term is: $$ t_n = (-1)^{n+1} \times \frac{1}{n} $$ 6. **Summary:** - To analyze alternating sequences, use the formula $a_n = (-1)^{n+1} \times c$. - To analyze sequences with decreasing magnitude like the bar graph, use $t_n = \frac{1}{n}$. - Combining both gives $t_n = (-1)^{n+1} \times \frac{1}{n}$. This explains the pattern of alternating signs and decreasing lengths in the bar graph.