Question

In winter, the temperature at a hill station from Monday to Friday is in A.P. the sum of the temperature of Monday, Tuesday and Wednesday is zero and the sum of the temperature of Thursday and Friday is 15. Find the temperature of each of the five days.
A. -3, 0, 3, 6, 9
B. -3, 3, 9, 15, 21
C. -2, 0, 2, 4, 6
D. 0, 3, 6, 9, 12

Answer 1

Hint: In the above question we will suppose the first term and common difference of the A.P, since the temperatures are in A.P from Monday to Friday. Then, we will write the temperature of each day using the formula of \[{{n}^{th}}\] term of an A.P given by,
\[{{t}_{n}}=a+\left( n-1 \right)d\]
Where 'a' is the first term and d is a common difference. To get started we have to consider \[\text{n}=\text{1}\] for Monday and compute \[{{t}_{n}}\] similarly all the terms can be found out.

Complete step-by-step answer:
We have been given that the temperature from Monday to Friday is in A.P. the sum of the temperature of Monday, Tuesday and Wednesday is zero and the sum of the temperature of Thursday and Friday is 15.
Let us suppose the temperature is in A.P having first term a and common difference d.
We know that \[{{n}^{th}}\] term of an A.P is given by,
\[{{t}_{n}}=a+\left( n-1 \right)d\]
On Monday, \[n=1\]
\[\begin{align}
  & \Rightarrow {{t}_{1}}=a+\left( 1-1 \right)d \\
 & \Rightarrow \text{ a} \\
\end{align}\]
On Tuesday, \[n=2\]
\[\begin{align}
  & \Rightarrow {{t}_{2}}=a+\left( 2-1 \right)d \\
 & \Rightarrow a+d \\
\end{align}\]
On Wednesday, \[n=3\]
\[\begin{align}
  & \Rightarrow {{t}_{3}}=a+\left( 3-1 \right)d \\
 & \Rightarrow a+2d \\
\end{align}\]
On Thursday, \[n=4\]
\[\begin{align}
  & \Rightarrow {{t}_{4}}=a+\left( 4-1 \right)d \\
 & \Rightarrow a+3d \\
\end{align}\]
On Friday, \[n=5\]
\[\begin{align}
  & \Rightarrow {{t}_{5}}=a+\left( 4-1 \right)d \\
 & \Rightarrow a+4d \\
\end{align}\]
According to question, the sum of temperature of Monday, Tuesday and Wednesday is zero.
\[\begin{align}
  & \Rightarrow a+\left( a+d \right)+\left( a+2d \right)=0 \\
 & \Rightarrow 3a+3d=0 \\
\end{align}\]
Having 3 as common, we get
\[\Rightarrow 3\left( a+d \right)=0\]
On dividing the equation by 3, we get
\[\Rightarrow a+d=0\]
\[\Rightarrow a=-d.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }.\text{ }\left( \text{1} \right)\]
Also, the sum of temperature of Thursday and Friday is 15.
\[\begin{align}
  & \Rightarrow a+3d+a+4d=15 \\
 & \Rightarrow 2a+7d=15 \\
\end{align}\]

On substituting a = -d using (1), we get
\[\begin{align}
  & \Rightarrow 2\left( -d \right)+7d=15 \\
 & \Rightarrow -2d+7d=15 \\
 & \Rightarrow 5d=15 \\
 & \Rightarrow d=\dfrac{15}{5}=3 \\
\end{align}\]
Using the equation (1), we get
\[a=-3\]
We have \[a=-3\text{ and d=3}\]
Temperature on Monday \[\Rightarrow a=-3\]
Temperature on Tuesday \[\Rightarrow a+d\]
\[\begin{align}
  & \Rightarrow -3+3 \\
 & \Rightarrow 0 \\
\end{align}\]
Temperature on Wednesday \[\Rightarrow a+2d\]
\[\begin{align}
  & \Rightarrow -3+2\times 3 \\
 & \Rightarrow -3+6 \\
 & \Rightarrow 3 \\
\end{align}\]
Temperature on Thursday \[\Rightarrow a+3d\]
\[\begin{align}
  & \Rightarrow -3+3\times 3 \\
 & \Rightarrow -3+9 \\
 & \Rightarrow 6 \\
\end{align}\]
Temperature on Friday \[\Rightarrow a+4d\]
\[\begin{align}
  & \Rightarrow -3+4\times 3 \\
 & \Rightarrow -3+12 \\
 & \Rightarrow 9 \\
\end{align}\]
So, the correct answer is “Option A”.

Note:Sometimes you get the answer by seeing option only, it is one of the questions you can get the answer by checking the options according to the given condition. It will save your time in competitive examinations. So, here we have option A and C which have the sum of temperature Monday, Tuesday and Wednesday as 0. Now the next condition is that the sum of temperatures on Thursday, Friday must be 15. In option A, we have 6 and 9 which add up to 15. But in option C, we have 4 and 6, which add up to 10. So clearly we get option A as the answer.Also, remember that in the question related to A.P and you have to find the first term or anything related with it then you should first suppose the variables that are not given and then solve it according to the given condition in the question.