nth Term of AP

The nth term of an AP is the term in the nth position from the beginning of the sequence. In arithmetic progressions, each term is obtained by adding a fixed value called the common difference to the previous term. In an AP, the first term is represented by ‘a’, and the common difference is d. 

For example, 2, 4, 6, 8, … here, the first term (a) is 2, the common difference (d) is 2(4 - 2 = 2). So, each term is formed by adding 2 to the previous term, for example, a15 = a14 + 2. 
 

Any term of an arithmetic progression can be found by adding the common difference to its previous term. The nth term of an AP is calculated by using the formula: 
an = a + (n - 1)d
where an is the nth term of the sequence
a is the first term
n is the index (position) of the term in the sequence
d is the common difference. 
 

To find any term of a sequence without knowing the previous term, we use the nth term of an AP formula. Any term in an AP sequence can be found using this formula. It is hard for students to calculate the 20th term by repeatedly adding the common difference. That is when we use the nth term formula, as we don't need to find the previous term. The nth term of an AP formula is: an = a + (n - 1)d. 
 

The sequence formed by adding the common difference to each term is called arithmetic progression. Now let’s see how to derive the nth term of an AP. 
Let’s consider the AP as:
a, a + d, a + 2d, a + 3d, …. 
Here, 1st term = a
Second term (a2) = a + d
Third term (a3) = (a + d) + d = a + 2d
Fourth term (a4) = (a + 2d) + d = a + 3d
Fifth term (a5) = (a + 3d) + d = a + 4d
…………..
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nth term(an)= a + (n - 1)d

So, the formula to find the nth term of an AP is: an = a + (n - 1)d

We have to calculate the nth term of an AP when the previous term is unknown. Let’s see how the formula is used to find the nth term of an AP.  

Step 1: Understanding the formula
The nth term of an AP formula: an = a + (n - 1)d. 
Where an is the nth term, 
a is the first term, 
n is the number of terms, 
d is the common difference. 

Step 2: Identifying the given sequence
From the given sequence, identify the values of a, n, and d.

Step 3: Substitute into the formula
Substitute the identified values into the formula to find the nth term. 

For example, find the 11th term of the sequence: 7, 11, 15, 19, ….
To find the nth term of an AP we use the formula: an = a + (n - 1)d
Now, let’s identify the values. Here, a = 7
d = 11 - 7 = 15 - 11 = 4
n = 11

a11 = 7+ (11 - 1)4
7 + 10 × 4 
= 47

So, the 11th term of the AP is 47.
 

The nth term of an AP is used in finance, engineering, scheduling, etc. In this section, we will learn a few applications of the nth term of AP. 

• In financial planning and savings, we use AP to predict the future savings amounts for budgeting or financial goals. For example, if we save $100 in the first month and then increase the savings by $20 each month, to find the amount saved in the 15th month, we can use the formula an = n + (n - 1)d. 

• In construction and engineering, we use the nth term of an AP to track productivity and set the timelines. For example, if a worker produces 10 units on the first day and increases the output by 3 units daily. To find the units produced on 15th day, we use the nth term of an AP formula. 

• Students can use the nth term of an AP to set a daily study goal. For example, if the students study 1 page on day 1, 2 pages on day 2, 3 pages on day 3, and so on. The sequence formed is 1, 2, 3, 4, … To find the number of pages studied on the 15th day, we can use the nth term formula.