Question

In winter, the temperatures at a hill station from Monday to Friday are in A.P. The sum of the temperatures from Monday to Wednesday is \[0^\circ \] , while the sum of the temperatures on Thursday and Friday is \[15^\circ \] . Find the temperatures on each weekday.

Answer 1

Hint: In this question, we need to find the temperature of each weekday. For solving this, we are going to need to assume the temperature of the middle day as the starting one. Then, we are going to denote the temperature of each day with respect to the middle day’s temperature plus or minus the assumed common difference times some natural number. Once we have the temperature of all the days, we are going to put in the values in the given equations in the question, calculate the assumed values, put them in the equation of each day and then we are going to have our answer.
Formula Used:
We are going to use the formula of arithmetic progression, which is:
 \[{a_n} = a + \left( {n - 1} \right)d\] , where \[{a_n}\] is the first term, \[a\] is the first term and \[d\] is the common difference.

Complete step-by-step answer:
The given days are Monday, Tuesday, Wednesday, Thursday, and Friday.
So, we take the middle day (Wednesday) as the base day and we are going to represent the other days with respect to Wednesday.
Now, let the temperature on Wednesday be \[a\] and since it forms an arithmetic progression, let the common difference be \[d\] .
So, the temperature on the other days is:
Monday: \[a - 2d\] Tuesday: \[a - d\] Wednesday: \[a\] Thursday: \[a + d\] Friday: \[a + 2d\]
Now, according two the first given information, we have:
 \[a - 2d + a - d + a = 0\]
or, \[3a - 3d = 0\] (i)
Then, with the second piece of information, we have:
 \[a + d + a + 2d = 15\]
or, \[2a + 3d = 15\] (ii)
Adding equations (i) and (ii),
 \[5a = 15\]
or, \[a = 3\]
Putting \[a = 3\] in equation (i) we get,
 \[d = 3\]
Hence, the temperature on each day is:
Monday: \[3 - 2 \times 3 = - 3^\circ \]
Tuesday: \[3 - 3 = 0^\circ \]
Wednesday: \[3^\circ \]
Thursday: \[3 + 3 = 6^\circ \]
and, Friday: \[3 + 6 = 9^\circ \]

Note: So, we saw that in solving questions like these, first we need to assume the first term of the sequence (represented by ‘a’), the common difference of the given arithmetic progression (represented by ‘d’). Then we represent all the given unknown values as first term plus or minus the common difference times some natural number. Then we put in the assumed values of the unknown quantities into the equations given in the question, calculate the assumed values, put them in the equation of each unknown value and then we are going to have our answer.