Question

Which of the following is the general formula of the arithmetic progression with the first term as ‘a’ and the difference as ‘d’?
A. \[a,\,a - d,\,a + 2d,\,a - 3d,\,...\]
B. \[a,\,a + d,\,a + 2d,\,a + 3d,\,...\]
C. \[a,\,a - d,\,a - 2d,\,a - 3d,\,...\]
D. \[a,\,a + 2d,\,a + 4d,\,a + 6d,\,...\]

Answer 1

Hint: According to the question, we should first understand what arithmetic progression is. Then only, we will be able to find out its formula with the first term as ‘a’ and the difference as ‘d’, based on the question.

Complete step-by-step solution:
We know that an arithmetic progression is a collection of numbers arranged in a proper sequence in which except the first term, all the other terms are obtained by adding a constant number or amount to the proceeding term.
This tells us that all the numbers differ from each other. All the numbers differ by the one next to them by a fixed number.
If we take an example of a sequence of numbers like, \[1,\,3,\,5,\,7,\,9,\,11,\,....\]. This sequence is an AP series. The two successive terms have a difference of a fixed value. If we take the two successive numbers \[5\,\text{ and }\,7\], then we will find their difference is \[2\,(7 - 5 = 2)\]. If we take another pair of successive numbers say \[9\,\text{ and }\,11\], then their difference is also \[2\,(11 - 9 = 2)\].
This shows us that this sequence is an AP series.
Now, we will check all the four options, and find out, which formula goes with the above concept. As we can see that all the options are starting with the term ‘a’. So, we have to just find out which option is having the difference as ‘d’.
If we check option A, then we see that there is no fixed value which is getting added. And, the second and the fourth term are getting subtracted. So, the difference will not come as ‘d’. Therefore, this option is incorrect.
If we check option B. then we will see that all the terms are getting added by a fixed value ‘d’. So, if we take the difference of any two successive terms, then their difference will be ‘d’. So, option B is correct.
If we check the third option, then we will see that all the terms are getting subtracted. So, we will not get the difference as ‘d’. Therefore, option C is incorrect.
If we check the last option, we will see that all the terms are getting added but not by a fixed amount. So, we will not get the difference as ‘d’. Therefore, option D is incorrect.

Note: When we look into our normal lives, we can find this arithmetic progression everywhere. For example, the days in a week, the months in a year, our roll numbers in class. All these things are categorized in Mathematics as progressions.