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. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag?

#### Solution

Given that, the number of flags = 27 and

Distance between each flag = 2 m.

Also, the flags are stored at the position of the middlemost flag

i.e., 14^{th} flag and Ruchi was given the responsibility of placing the flags.

Ruchi kept her books, where the flags were stored

i.e., 14^{th} flag and she coluld carry only one flag at a time.

Let she placed 13 flags into her left position from middlemost flag

i.e., 14^{th} flag. For placing second flag and return her initial position distance travelled = 2 + 2 = 4 m.

Similarly, for placing third flag and return her initial position,

Distance travelled = 4 + 4 = 8m

For placing fourth flag and return her initial position,

Distance travelled = 6 + 6 = 12 m

For placing fourteenth flag and return her initial position,

Distance travelled = 26 + 26 = 52 m

Proceed same manner into her right position from middlemost flag i.e., 14^{th} flag.

Total distance travelled in that case = 52 m

Also, when Ruchi placed the last flag she return his middlemost position and collect her books.

This distance also included in placed the flag.

So, these distance form a series 4 + 8 + 12 + 16 + …. + 52 ......[For left]

And 4 + 8 + 12 + 16 + … + 52 ....[For right]

∴ Total distance covered by Ruchi for placing these flags

= 2 × (4 + 8 + 12 + … + 52)

= `2 xx [13/2 {2 xx 4 + (13 - 1) xx (8 - 4)}]`

∵ Sum of n terms of an AP, `S_n = n/2[2a + (n - 1)d]`

= `2 xx 13/2 (8 + 12 xx 4)` ......[ ∵ Both sides of number of flags i.e., n = 13]

= 2 × [13(4 + 12 × 2)]

= 2 × 13(4 + 24)

= 2 × 13 × 28

= 728 m

Hence, the required distance is 728 m in which she did cover in completing this job and returning back to collect her books.

Now, the maximum distance she travelled carrying a flag = Distance travelled by Ruchi during placing the 14^{th} flag in her left position or 27^{th} flag in her right position

= (2 + 2 + 2 + … + 13 times)

= 2 × 13

= 26 m

Hence the required maximum distance she travelled carrying a flag is 26 m.