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(A) Given,total number of flags $= 27$. The flags are placed at intervals of $2 \,m$. The middle-most flag is the $14^{th}$ flag. Ruchi starts from th

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$728, 26$

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(A) Given,total number of flags $= 27$. The flags are placed at intervals of $2 \,m$. The middle-most flag is the $14^{th}$ flag. Ruchi starts from the $14^{th}$ flag position.For placing flags on the left side: There are $13$ flags to the left. To place the $1^{st}$ flag (at $2 \,m$ from center),she travels $2 \,m$ and returns $2 \,m$ (Total $4 \,m$). To place the $2^{nd}$ flag (at $4 \,m$ from center),she travels $4 \,m$ and returns $4 \,m$ (Total $8 \,m$). This continues up to the $13^{th}$ flag (at $26 \,m$ from center),where she travels $26 \,m$ and returns $26 \,m$ (Total $52 \,m$).The total distance for the left side is $4 + 8 + 12 + \dots + 52$. This is an Arithmetic Progression $(AP)$ with $a = 4$,$d = 4$,and $n = 13$.Sum $= \frac{n}{2} [2a + (n-1)d] = \frac{13}{2} [2(4) + (12)(4)] = \frac{13}{2} [8 + 48] = \frac{13}{2} [56] = 13 	imes 28 = 364 \,m$.Similarly,for the right side,the total distance is also $364 \,m$.Total distance covered $= 364 + 364 = 728 \,m$.The maximum distance travelled carrying a flag is the distance to the furthest flag from the center,which is $26 \,m$.

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