Arithmetic Progression

Arithmetic Progression – Definition and formulas

An arithmetic progression (AP) is a sequence, where the difference between any two consecutive numbers is constant. For example, consider the 5 tables. We have 5, 10, 15, 20, and so on. The difference between any two successive numbers in this series is always 5. So this is an arithmetic progression with the common difference of 5. 

We can find arithmetic progression series all around us. We can find AP in a taxi fare where the fare per additional kilometer increases by a fixed amount or the fixed percentage of interest in a fixed deposit. Let us learn in detail about the various terms and formulas in arithmetic progressions here. 

Terms in an Arithmetic Progression:

An arithmetic progression can be specified by two terms which are the first term, and the common difference.

The first term, as the name suggests, is the first term of the arithmetic sequence and is denoted generally as a. In an AP containing the terms 7, 11, 15, 19, 23 …, the first term a = 7. 

The common difference is the constant number which is the difference between any two consecutive terms in an AP. We can add the common difference to the previous term to arrive at the next term in the series. It is generally denoted as d. 

Common difference d = a_n – a_n-1

Where

d    = common difference

a_n = nth term in an Arithmetic Progression

a_n-1 = (n-1)th term in an Arithmetic Progression

In the above mentioned series, the common difference d is (11-7) = (15-11) = 4

The general form of an arithmetic progression is a, a+d, a+2d, a+3d, ….. a+(n-1)d

Where

a= first term

d= common difference and

n= number of terms

Formulas in Arithmetic Progression: 

nth term of an AP 

In an arithmetic progression where the first term and the common difference is known, the nth term of the series can be calculated using the formula T_n = ((a+(n-1))d 

 If we want to calculate the nth term from the end of the AP, then we use the formula 
_Tn=( l-(n-1))d 

where l is the last term in an AP

Number of terms in an AP

When the first and last terms of an AP and the common difference is known, then we can calculate the number of terms in the arithmetic progression using the formula n=(l-a)d+1

Sum of first n terms in an AP

When we know the first term and the common difference in the AP, the formula for calculating the sum of an arithmetic progression is S= n[2a+(n-1)d]/2.

When we know the first and the last term, the sum of n terms in an AP is S=n(a+l)/2

Where a is the first term and l is the last term.

Sum of an infinite arithmetic progression

If the common difference in the AP is greater than 0, the sum of the AP tends to +∞ , and the sum of an AP where the common difference is less than 0, tends to -∞ .

Formula summary

Where 

a= first term

d= common difference and

n= number of terms

         l= last term in an AP

Solved Examples:

1. Find the sum of the natural numbers from 501 to 1000. 

We can calculate the sum  of the numbers in an Arithmetic Progression where the first and the last terms are known using the formula S= n(a+l))/2

We need to find the sum of the natural numbers from 501 to 1000, where the first term is 501, the last term is 1000 and the common difference is 1. The number of terms in this arithmetic progression is 500.

Substituting the above figures in the formula, we get

S50=n(a+l)/2

S50=500(501+1000)/2

S50=7,50,5002

S50=3,75,250

The sum of the natural numbers from 501 to 1000 is 3,75,250.

2. In a building, the first floor has 69 rooms, the second floor has 64 rooms, the third floor has 59 rooms, and so on. The final floor has 9 rooms. Find the total number of floors in the building. 

Solution:

The number of rooms in this building is 69, 64, 59, ….. 9. We can see that this forms an arithmetic progression with the first term a = 69, common difference d = 64-69 = -5 and the nth term is 9. 

We know that the nth term of an AP an = a+ (n-1)/d

Substituting the given values in this formula, we get

  9 = 69 + (n-1)(-5)

-60 = -5n + 5.

65  = 5n

  n  = 13.

So, the building has 13 floors.

3. Find the sum of the first 20 terms of the AP 4, 11, 18, 25 ….

Solution: 

 This is an AP with the first term a = 4 and the common difference d = 11-4 = 7.

The sum of first n terms of an AP Sn = n[2a+(n-1)d]/2

S20 = 20[2✕4+(20-1)7]/2

S_{20} =\frac{20[2*4+(20-1)7]}{2}

S20 = 20[141]/2

S20 = 1,410