5.2. Arithmetic Progressions

Consider the following lists of numbers :

(i) 1, 2, 3, 4, . . .

(ii) 100, 70, 40, 10, . . .

(iii) –3, –2, –1, 0, . . .

(iv) 3, 3, 3, 3, . . .

(v) –1.0, –1.5, –2.0, –2.5, . . .

Each of the numbers in the list is called a term. Given a term, can you write the next term in each of the lists above? If so, how will you write it? Perhaps by following a pattern or rule. Let us observe and write the rule.

An Arithmetic Progression (AP) consists of a sequence of numbers where the difference between consecutive terms is constant. This fixed difference is called the common difference, denoted as 'd'. Each list presented shows a pattern in which you can identify the rule to find the next term.

• For the list (i), each term increases by 1, so it's 1 + 1 = 2, and so forth.
• In list (ii), each term decreases by 30, leading to 100 - 30 = 70.
• List (iii) illustrates that each term increases by 1 starting from -3, and so on for the rest. The same concept applies to lists (iv) and (v). All these examples depict an arithmetic sequence, highlighting that when moving from one term to the next, you can determine future terms using the common difference.

Think of climbing stairs. Each step is equidistant apart, similar to how terms in an arithmetic sequence are spaced by a consistent amount, the common difference. Just like knowing the height of the first step allows you to easily find the height of the following steps, knowing the first term and the common difference helps you find subsequent terms in an AP.

In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form an Arithmetic Progression (AP).

So, an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term. This fixed number is called the common difference of the AP. Remember that it can be positive, negative or zero.

An Arithmetic Progression is fundamentally a sequence where consecutive terms are generated by adding the same value, known as the common difference (d), to the previous term. This can manifest in various forms:

• If d is positive, the terms increase.
• If d is negative, the terms decrease.
• If d equals zero, all terms remain the same.

For example, if the first term (a) is 5 and d is 3, the sequence goes 5, 8, 11, 14,... If d is -2, starting from 5 yields 5, 3, 1, -1,... This emphasizes the versatility of APs.

Consider a savings plan where you deposit a fixed amount every month. If you deposit $50 each month, your account balance follows an arithmetic progression:$50, $100,$150, $200, and so on. The common difference is the amount you add, which helps you predict your total savings at any point in time.

Let us denote the first term of an AP by a , second term by a , . . ., nth term by a and the common difference by d. Then the AP becomes a , a , a , . . ., a . So, a – a = a – a = . . . = a – a = d.

To denote the terms in an arithmetic progression clearly, we label the first term as a₁, the second term as a₂, and the nth term as aₙ. The common difference d can be identified between any two consecutive terms:

• d = a₂ - a₁
• d = a₃ - a₂
• This pattern continues for any terms in the sequence. This relationship illustrates how each term can be derived from its predecessor using the common difference.

Imagine organizing a baking schedule where every week you make three more cookies than the previous week. If your first week you bake 10 cookies, your second week would be 10 + 3 = 13, leading to a clear AP: 10, 13, 16, 19,... Here, 10 is your first term, and 3 is the common difference.

Some more examples of AP are: (a) The heights ( in cm ) of some students of a school standing in a queue in the morning assembly are 147 , 148, 149, . . ., 157. (b) The minimum temperatures ( in degree celsius ) recorded for a week in the month of January in a city, arranged in ascending order are – 3.1, – 3.0, – 2.9, – 2.8, – 2.7, – 2.6, – 2.5 (c) The balance money ( in ) after paying 5 % of the total loan of 1000 every month is 950, 900, 850, 800, . . ., 50. (d) The cash prizes ( in ) given by a school to the toppers of Classes I to XII are, respectively, 200, 250, 300, 350, . . ., 750. (e) The total savings (in) after every month for 10 months when ` 50 are saved each month are 50, 100, 150, 200, 250, 300, 350, 400, 450, 500.

In the examples listed, each scenario shows a distinct arithmetic progression where a fixed difference contributes to the creation of subsequent terms:

• (a) Height increases by 1 cm for each student.
• (b) Temperature rises by 0.1°C each day.
• (c) The loan balance decreases by 50 ₹ each month.
• (d) Each cash prize increases by 50 ₹ with each class.
• (e) Savings increase by a steady amount of 50 ₹ monthly. Understanding these examples provides insight into various real-world applications of AP.

Picture a hotel with a rising price per night where the cost increases by a fixed amount (such as $20) each week. If the price starts at$100 in the first week, it becomes $120 in the second,$140 in the third, and so on. This scenario represents an arithmetic progression where you can easily predict how much a stay will cost in subsequent weeks!

It is left as an exercise for you to explain why each of the lists above is an AP. You can see that a, a + d, a + 2d, a + 3d, . . . represents an arithmetic progression where a is the first term and d the common difference. This is called the general form of an AP.

The definition of an arithmetic progression can be summarized into a formula: a, a + d, a + 2d, a + 3d,... where a is the initial term, and d is the consistent difference that can be either positive, negative, or zero. By analyzing the lists presented, you can see how to apply the definition and recognize whether a number sequence is an AP based on this pattern.

Think of the layers of cake in a tiered cake: if you add a consistent height for each layer, the overall structure becomes an arithmetic progression by height. If each layer is 2 inches taller than the one beneath, starting at 4 inches, your layers are 4, 6, 8, 10,... consistently rising as an arithmetic progression!

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