The two particles will meet after 50 seconds
Distance traveled by the first particle = 1225 cm
Distance traveled by the second particle = 625 cm
Explanation:For the first partiicle
The sequence representing the distance traveled after every 2 seconds is:
97, 93, 89, .........
The common difference, d = 93 - 97 = 89 - 93
d = -4
The distance traveled in the first two seconds, a = 97
Let the number of terms be n
This is an arithmetic sequence, and the distance traveled after n 2 seconds will be:
[tex]\begin{gathered} T_n=\text{ a + (n-1)d} \\ T_n=\text{ 97 + (n-1)(-4)} \\ T_n=\text{ 97-4n+4} \\ T_n=\text{ 101-4n} \end{gathered}[/tex]For the second particle
The sequence representing the distance traveled every 2 seconds is:
49, 47, 45,......
The common difference, d = 47-49 = 45 - 47 = -2
d = -2
The distance traveled in the first two seconds is a = 49
The nth term of the sequence is:
[tex]\begin{gathered} T_n=\text{ a + (n-1)d} \\ T_n=49+(n-1)(-2) \\ T_n=\text{ 49-2n+2} \\ T_n=\text{ 51-2n} \end{gathered}[/tex]When the two particles collide, the two equations can be equated
[tex]\begin{gathered} 101-4n\text{ = 51 - 2n} \\ 4n\text{ - 2n = 101-5}1 \\ 2n\text{ = }50 \\ n=\frac{50}{2} \\ n=25 \end{gathered}[/tex]These two particles will collide after 25 distances, and since each distance takes 2 seconds, the time taken for the particles to meet will be:
t = 2n
t = 2(25)
t = 50
The particles will meet after 50 seconds
Sum of n terms of an Arithmetic Sequence is:
[tex]S_n=\text{ }\frac{n}{2}\lbrack2a+(n-1)d\rbrack[/tex]For the first particle:
a = 97, d = -4, n = 25
[tex]\begin{gathered} S_{25}=\text{ }\frac{25}{2}\lbrack2(97)+(25-1)(-4)\rbrack \\ S_{25}=\text{ 12.5(}194-96) \\ S_{25}=\text{ }1225 \end{gathered}[/tex]The distance traveled by the first particle = 1225 cm
For the second particle:
a = 49, d = -2, n = 25
[tex]\begin{gathered} S_{25}=\text{ }\frac{25}{2}\lbrack2(49)+(25-1)(-2)\rbrack \\ S_{25}=\text{ 12.5(}98-48) \\ S_{25}=12.5(50) \\ S_{25}=\text{ }625 \end{gathered}[/tex]The distance traveled by the second particle = 625 cm
