Question

A train can travel 200 m in the first hour, 400 m the next hour, 600 m the third hour, and so on in an arithmetic sequence. What is the total distance the train travels in 5 hours?
(a) 2,000
(b) 3,000
(c) 4,000
(d) 5,000

Answer 1

Hint: In this question, first you need to arrange and form the sequence in an arithmetic sequence, which is (a, a + d, a + 2d, …) and so on. Now, you need to find the value of ‘a’ and ‘d’ and find the distance the train travels the fourth and fifth hour by using the formula, ${{T}_{n}}=\,a+\left( n-1 \right)d$. Now add all the distance the train covered in the respective hours from 1 to 5.

Complete step-by-step solution:
Let us first determine the sequence according to the question,
200, 400, 600, …….
The sequence is in the form of an arithmetic sequence, hence compare the above sequence with
a, a + d, a + 2d, a + 3d, …….
After comparing, the above two sequences we get, a = 200, d = 200
Also if you notice, ${{T}_{1}}=200$, ${{T}_{2}}=400$, ${{T}_{3}}=600$ with which we can prove ${{T}_{n}}=\,a+\left( n-1 \right)d$, where $n$is the number of hours taken by the train. So, let us find the value of ${{T}_{4}},\,{{T}_{5}}$.
For ${{T}_{4}}$, substitute the values a = 200, d = 200 and n = 4
Using the formula for ${{T}_{n}}$, find the value of ${{T}_{4}}$, we get
$\begin{align}
  & {{T}_{4}}=a+\left( n-1 \right)d \\
 & =200+\left( 4-1 \right)200 \\
 & =200+\left( 3 \right)200 \\
 & =200+600 \\
 & =800
\end{align}$
So, in the 4th hour the train travels about 800 m.
For ${{T}_{5}}$, substitute the values a = 200, d = 200, n = 5
Similarly, using the formula for ${{T}_{n}}$, find the value of ${{T}_{5}}$, we get
$\begin{align}
  & {{T}_{5}}=a+\left( n-1 \right)d \\
 & =200+\left( 5-1 \right)200 \\
 & =200+\left( 4 \right)200 \\
 & =200+800 \\
 & =1000
\end{align}$
So, in the 5th hour the train travels about 1000 m.
Now, you have to find what is the total distance covers in 5 hours, to find this, you need to add all the values, that is,
$\begin{align}
  & {{T}_{total}}={{T}_{1}}+{{T}_{2}}+{{T}_{3}}+{{T}_{4}}+{{T}_{5}} \\
 & =200+400+600+800+1000 \\
 & =3000
\end{align}$
Hence, the total distance travelled by the train in 5 hours is 3000 m.

Note: Here, the most important is to determine the sequence. Sometimes the nature of the sequence won’t be given, it is then important to find the sequence by arranging them from smaller to higher value and comparing them with ‘a, a + d, a + 2d, ……’. You can also use the formula to easily solve by using Sum of first n terms, ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$.