Case study:
The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. A total of 27 flags have to be fixed at intervals of every 3 m. All the flags are kept at the position of the middle most flag. Rohan was given the responsibility of placing the flags on both the sides of the middle most flag. Rohan could carry only one flag at a time. So he carries one flag from the middle most point, places the flag at the designated place and comes back to the middle most point. He continues this for all the flags on both sides.

Based on the above information, answer the following questions:

(i) The distance covered by Rohan in placing the flags on one side of the middle most flag forms a sequence. Identify the type of sequence by writing at least the first 4 terms of the sequence.

(ii) Find the distance covered by Rohan in placing the 10^th flag and coming back to the middle most flag.

(iii) Find the total distance travelled by Rohan in fixing all the 27 flags.

(i) There are 27 flags in total. The middle-most flag is the 14th flag.

This means there are 13 flags to the left and 13 flags to the right of the starting point.

The first flag on either side is 3 meters away. Rohan travels 3 m to place it and 3 m to return.

The second flag is 6 m away. Rohan travels 6 m to place it and 6 m to return.

The first 4 terms (round-trip distances) are :

1st flag = 2 × 3 = 6 m

2nd flag = 2 × 6 = 12 m

3rd flag = 2 × 9 = 18 m

4th flag = 2 × 12 = 24 m

The above terms form an A.P. with common difference = 6.

Hence, the required sequence is an A.P.

(ii) By formula,

a_n = a + (n - 1)d

a₁₀ = 6 + 6(10 - 1)

= 6 + 6 × 9

= 6 + 54

= 60 m.

Hence, the distance covered by Rohan in placing the 10^th flag and returning back to the middle most flag = 60 m.

(iii) By formula,

S_n = $\dfrac{n}{2}[2a + (n - 1)d]$

On each side 13 flags are placed.

$S_{13} = \dfrac{13}{2}[2 \times 6 + (13 - 1) \times 6] \\[1em] = \dfrac{13}{2} \times [12 + 12 \times 6] \\[1em] = \dfrac{13}{2} \times [12 + 72] \\[1em] = \dfrac{13}{2} \times 84 \\[1em] = 13 \times 42 \\[1em] = 546 \text{ m}.$

Thus, distance covered on one side = 546 m.

Total distance = 2 × 546 = 1092 m.

Hence, total distance travelled by Rohan in fixing all the 27 flags = 1092 m.