7. The two trains $$ A $$ and $$ B $$ travel in opposite directions on two parallel tracks, heading each other. Train $$ A $$ travelled 1 m in the $$ 1^{ ext {st }} $$ second, 3 m in the $$ 2^{ ext {nd }} $$ second 5 m in the $$ 3^{ ext {rd }} $$ and so on while train $$ B $$ travelled 11 m in the $$ 1^{ ext {st }} $$ second, 12 m in the $$ 2^{ ext {nd }} $$ second 13 m in the $$ 3^{ ext {rd }} $$ and so on. (a) The distances the two trains travelled in the $$ n^{ ext {th }} $$ second are equal. Find the value of $$ n $$. (b) The two trains met each other at the end of the $$ n^{ ext {th }} $$ second. Find the distance between the two trains at the beginning.

Question

7. The two trains $$ A $$ and $$ B $$ travel in opposite directions on two parallel tracks, heading each other. Train $$ A $$ travelled 1 m in the $$ 1^{	ext {st }} $$ second, 3 m in the $$ 2^{	ext {nd }} $$ second 5 m in the $$ 3^{	ext {rd }} $$ and so on while train $$ B $$ travelled 11 m in the $$ 1^{	ext {st }} $$ second, 12 m in the $$ 2^{	ext {nd }} $$ second 13 m in the $$ 3^{	ext {rd }} $$ and so on. (a) The distances the two trains travelled in the $$ n^{	ext {th }} $$ second are equal. Find the value of $$ n $$. (b) The two trains met each other at the end of the $$ n^{	ext {th }} $$ second. Find the distance between the two trains at the beginning.

UpStudy AI · Accepted Answer

**(a) Finding $$ n $$ such that the distances travelled in the $$ n^{\text{th}} $$ second are equal**

For train $$ A $$, the distance in the $$ n^{\text{th}} $$ second is:
$$
a_n = 2n - 1
$$

For train $$ B $$, the distance in the $$ n^{\text{th}} $$ second is:
$$
b_n = 10 + n
$$

Setting these equal:
$$
2n - 1 = 10 + n
$$

Subtract $$ n $$ from both sides:
$$
n - 1 = 10
$$

Add $$ 1 $$ to both sides:
$$
n = 11
$$

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**(b) Finding the initial distance between the two trains**

The trains meet at the end of the $$ 11^{\text{th}} $$ second (from part (a)). To find the initial distance, we sum the distances each travelled in the $$ 11 $$ seconds.

*For train $$ A $$:*  
The distances form the sequence $$ 1, 3, 5, \ldots $$ and the sum of the first $$ n $$ odd numbers is:
$$
S_A = n^2
$$
For $$ n = 11 $$:
$$
S_A = 11^2 = 121
$$

*For train $$ B $$:*  
The distances form an arithmetic sequence with first term $$ 11 $$ and $$ n^{\text{th}} $$ term $$ (10+n) $$. The sum of an arithmetic series is given by:
$$
S_B = \frac{n}{2} \left( \text{first term} + \text{last term} \right) = \frac{n}{2} \left( 11 + (10+n) \right)
$$
For $$ n = 11 $$:
$$
S_B = \frac{11}{2} \left( 11 + (10+11) \right) = \frac{11}{2} \left( 11 + 21 \right) = \frac{11}{2} \times 32 = 11 \times 16 = 176
$$

The total initial distance between the trains is the sum of $$ S_A $$ and $$ S_B $$:
$$
D = S_A + S_B = 121 + 176 = 297
$$

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**Final Answers:**

(a) $$ n = 11 $$  
(b) The initial distance between the two trains is $$ 297 $$ meters.
(a) The trains are equal in the 11th second. (b) The initial distance between the trains is 297 meters.

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