The grade 10 students formed a human pyramid in the school field demonstration. The first row had one student, and the third row had five students. If there were eight rows, how many students were in the last row?



Answer :

Let's solve this question step by step.

1. Identify the pattern:
- We know that the first row has 1 student.
- We know that the third row has 5 students.

2. Determine the increment in the number of students per row:
- We observe that from the first row to the third row, the increase in the number of students is from 1 to 5.
- This difference can be noted as follows:
[tex]\[ \text{Difference from Row 1 to Row 3} = 5 - 1 = 4 \text{ students} \][/tex]
- This difference occurs over two intervals (from Row 1 to Row 2 and from Row 2 to Row 3), so the increment per row is:
[tex]\[ \text{Increment per row} = \frac{4 \text{ students}}{2 \text{ intervals}} = 2 \text{ students per row} \][/tex]

3. Find the number of students in each row by adding the increment:
- First row: 1 student (given)
- Second row:
[tex]\[ 1 + 2 = 3 \text{ students} \][/tex]
- Third row:
[tex]\[ 3 + 2 = 5 \text{ students} \][/tex]
- We see that this pattern holds.

4. Apply this pattern to find the number of students in the last (8th) row:
- We can generalize the pattern in terms of rows. If [tex]\( n \)[/tex] is the row number, then the number of students in the [tex]\( n \)[/tex]-th row can be written as:
[tex]\[ \text{Students in } n \text{th row} = 1 + (2 \times (n - 1)) \][/tex]

5. Substitute for the 8th row:
- Using [tex]\( n = 8 \)[/tex]:
[tex]\[ \text{Students in 8th row} = 1 + (2 \times (8 - 1)) = 1 + (2 \times 7) = 1 + 14 = 15 \text{ students} \][/tex]

Thus, the number of students in the last (8th) row is [tex]\( 15 \)[/tex].