Question

Find the term ${1^2},{5^2},{7^2},73.....$ is it an AP? If yes, then what is its common difference?
A.No
B.Yes, $d = 15$
C.Yes, $d = 24$
D.Yes, $d = 25$

Answer 1

Here, we are required to prove whether the given sequence is an Arithmetic Progression or not. We will prove that this sequence is an Arithmetic progression or A.P. by subtracting each term and its preceding term, this will give us the ‘common difference’. If we will get the same common difference for two consecutive terms, then this sequence will be an A.P. and our common difference will be the number by which each term differs from the previous one.

Complete step-by-step answer:
The sequence is ${1^2},{5^2},{7^2},73.....$.
Now, an Arithmetic Progression or A.P. is a sequence in which the difference between two consecutive terms is the same.
In the given sequence, the difference between the second and the first term is:
Difference $= {5^2} - {1^2} = 25 - 1 = 24$
Also, the difference between the third and the second term is:
Difference $= {7^2} - {5^2} = 49 - 25 = 24$
Again, the difference between the fourth and the third term is:
Difference $= 73 - {7^2} = 73 - 49 = 24$
Now we can see that the difference between the terms and their preceding terms is the same, i.e. 24.
So, we can conclude that the given sequence is an Arithmetic Progression or A.P. whose common difference, $d = 24$.
Therefore, option C is the correct answer.

Note: Arithmetic progressions are also used in real life such as adding the same amount as our pocket money in our money bank. Since, we add the same amount each time, that amount or that pocket money will become our common difference in this case. Similarly, we hire a taxi, we are charged an initial rate and then rate per kilometer. That rate per kilometer becomes our common difference and each addition gives us an A.P. Hence, this is used in our day to day life.