nth Term of AP
The n^{th} term of AP (Arithmetic Progression) is used to find any term of it. Usually, a term of AP is obtained by adding the common difference of AP to its previous term. But using the n^{th} term of AP formula, we can find any term of AP without needing to know its previous term.
Let us see what is the formula for n^{th} term of AP and how to derive it. We also use the formula and solve example problems.
1.  What is nth Term of AP? 
2.  nth Term of AP Formula 
3.  FAQs on nth Term of AP 
What is nth Term of AP?
The n^{th} term of AP is the term that is present in the n^{th} position from the first (left side) of an arithmetic progression. An arithmetic progression can be defined as a sequence where the differences between every two consecutive terms are the same. Consider the following AP: 2, 5, 8, .... Here, the first term a of this AP is T_{1} = 2, the second term is T_{2} = 5, the third term is T_{3 }= 8, and so on. We can clearly see that the common difference of this AP is 5  2 = 3. So if we want to find the fourth term, we can just add the common difference, 3 to the third term. i.e., T_{4} = T_{3} + 3 = 8 + 3 = 11. Similarly, the fifth term, T_{5} = 11 + 3 = 14. But what if we want the terms like T_{20}, T_{45}, T_{90}, T_{200}, etc? Applying this technique of adding the common difference to the previous term to get the current term would be difficult in such cases. The n^{th} term of AP formula simplifies this process.
nth Term of AP Formula
As we have discussed earlier, we cannot evaluate each and every term of the AP to determine these specific terms (such as T_{20}, T_{45}, T_{90}, T_{200}, etc). Instead, we must develop a relationship that enables us to find the n^{th} term for any value of n. To do that, consider an AP whose first term is 'a' and the common difference is 'd'. Then its terms are:
T_{1} = a
T_{2} = a + d
T_{3} = a + d + d = a + 2d
T_{4} = a + 2d + d = a + 3d
T_{5} = a + 3d + d = a + 4d
T_{6} = a + 4d + d = a + 5d
What pattern do you observe? If we have to calculate the sixth term, for example, then we have to add five times d (common difference) to the first term a. Similarly, if we have to calculate the n^{th} term, how many times will we add d to a? The answer should be easy: one less than n (or) n  1.
Thus, the formula of n^{th} term of ap is,
 T_{n} = a + (n  1)d
This relation helps us calculate any term of an AP, given its first term (a) and its common difference (d).
Thus, for the AP that is mentioned in the previous section (2, 5, 8, ...), we have:
 T_{20} = 2 + (20  1) 3 = 2 + 57 = 59
 T_{45} = 2 + (45  1) 3 = 2 + 132 = 134
 T_{90} = 2 + (90  1) 3 = 2 + 267 = 269
 T_{200} = 2 + (200  1) 3 = 2 + 597 = 599
Let us look into more examples.
Example 1: What is the 11^{th} term for the given arithmetic progression 2, 6, 10, 14, 18, ...?
Solution:
In the given arithmetic progression,
 First term = a = 2
 Common difference = d = 4
Term to be found is 11^{th} term. So n = 11.
Hence, the 11^{th} term for the given progression is,
T_{n} = a + (n  1)d
T_{11} = 2 + (11  1)4 = 2 + 40 = 42.
Thus, the 11^{th} term of given AP is 42.
Example 2: If the 5^{th} term of an AP is 40 with a common difference of 6. Find out the arithmetic progression.
Solution:
The given values for the AP are,
 Fifth term = T_{5} = 40
 Common difference = d = 6
The fifth term can be written as,
T_{5} = a + (5  1)6 = a + 24 = 40
⟹ a = 40  24 = 16
Hence, the arithmetic progression is,
T_{1} = a = 16
T_{2} = a + d = 16 + 6 = 22
T_{3} = a + 2d = 16 + (2)(6) = 28
T_{4} = a + 3d = 16 + (3)(6) = 34
T_{5} = a + 4d = 16 + (4)(6) = 40
T_{6} = a + 5d = 16 + (5)(6) = 46
The arithmetic progression is, 16, 22, 28, 34, 40, 46, ....
Tips and Tricks on nth Term of AP:
 The common difference doesn't need to be positive always.
 The common difference of an AP is the difference between any term and its previous term taken in this order.
 As per the formula of n^{th} term of ap, the first term can be found by using a = T_{n}  (n  1)d.
☛ Related Topics:
nth Term of AP Examples

Example 1: Find the 20^{th} term of an AP whose 3^{rd} term is 5 and 7^{th} term is 13?
Solution:
It is given that T_{3} = 5 and T_{7} = 13.
Using the n^{th} term of AP formula,
a + 2d = 5 ... (1)
a + 6d = 13 ... (2)
Subtracting (1) from (2):
4d = 8 ⟹ d = 2.
Substituting this in (1):
a + 2(2) = 5 ⟹ a = 1.
Now, the 20^{th} term is:
T_{20} = a + (20  1) d = 1 + (19) (2) = 39.
Answer: The 20^{th} term of AP is 39.

Example 2: How many threedigit numbers are divisible by 3?
Solution:
The smallest threedigit number which is divisible by 3 is 102 and the largest threedigit number divisible by 3 is 999.To find the number of terms in the following AP: 102, 105, 108,..,999, we will take 999 be the n^{th} term of AP, it can be seen that a is equal to 102, and d is equal to 3.
Then:
T_{n} = a + (n  1)d
999 = 102 + (n  1)3
3(n  1) = 999  102 = 897
n  1 = 897 / 3
n  1 = 299
n = 300
Answer: There are 300 threedigit numbers that are divisible by 3.

Example 3: Determine whether 301 is a term of the AP 7, 11, 15, 19,...
Solution:
In the given AP, the first term is a = 7 and the common difference is d = 4.
Let us assume that 301 is the n^{th} term of AP. Then:
T_{n} = a + (n  1)d
301 = 7 + (n  1) 4
301 = 7 + 4n  4
301 = 4n + 3
298 = 4n
n = 74.5
But 'n' must be an integer. Hence 301 cannot be a term of the given AP.
Answer: 301 cannot be a term of the given AP.
FAQs on nth Term of AP
How Do You Find the nth Term of AP?
To find the n^{th} Term of AP,
 Find the first term of the AP and label it as "a".
 Find the common difference of the AP and label it as "d".
 Find "n" depending on which term we have to find.
 Substitute all these values in the formula T_{n} = a + (n  1)d. Here, T_{n} means the n^{th} term.
What is the Formula For the nth Term of an AP?
The formula for the n^{th} term of an AP is, T_{n} = a + (n  1)d, where:
 'a' is the first term of the AP and
 'd' is the common difference of the AP.
What is nth Term From Last of an AP?
Assume that there are 'm' terms in an AP (Arithmetic Progression) in total whose first term is 'a' and the common difference is 'd'. Then the formula for the n^{th} term from the last of AP (its position from first would be (mn+1)^{th} position) is: T_{mn+1 }= a + (mn+11)d = a + (m  n) d.
How Do I Find the First Term From nth Term of AP?
If the n^{th} term of an AP is given, say T_{n} = 2n + 7, then to find its first term, just substitute n = 1 in it. Then we get T_{1} = 2(1) + 7 = 9, is the first term of the AP.
How Do I Find the Common Difference From nth Term of AP?
If the n^{th} term of an AP is given, say T_{n} = 2n + 7, then to find its common difference, just find T_{1} and T_{2} and then do T_{2}  T_{1}.
T_{2} = 2(2) + 7 = 11.
T_{1} = 2(1) + 7 = 9
The common difference, d = T_{2}  T_{1} = 11  9 = 2. The easy way of finding the common difference from the n^{th} term is, just identifying the coefficient of n. In the above example, the coefficient of n is 2 and hence it is the common difference.
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