1. Introduction
The nth term of an AP tells us the value of any term in the sequence without writing all the previous terms. It is like a shortcut to jump directly to the required term.
If an AP is: \(a, a+d, a+2d, a+3d, ...\), the nth term gives the general form of the pattern.
2. Formula for the nth Term
The formula for the nth term of an AP is:
\(a_n = a + (n - 1)d\)
Where:
- \(a_n\): nth term
- \(a\): first term
- \(d\): common difference
- \(n\): term number
This formula works for any AP, whether it is increasing, decreasing, or constant.
3. Why the Formula Works
In an AP, every term increases or decreases by \(d\). So:
- 2nd term = \(a + d\)
- 3rd term = \(a + 2d\)
- 4th term = \(a + 3d\)
The nth term will have \((n−1)\) multiples of \(d\), which leads to:
\(a_n = a + (n - 1)d\)
4. Finding Terms Using the nth Term Formula
You can find any term of an AP using the formula as long as you know the first term and the common difference.
4.1. Example 1: Find 10th Term
AP: 3, 7, 11, 15, ...
Here, \(a = 3\), \(d = 4\)
10th term → \(a_{10} = 3 + 9(4) = 39\)
4.2. Example 2: Find 25th Term
AP: 12, 10, 8, 6, ...
\(a = 12, d = -2\)
\(a_{25} = 12 + 24(-2) = -36\)
5. Finding the Position of a Term (Solving for n)
Sometimes we know a term and want to find its position in the AP. Rearrange the formula:
\(a_n = a + (n-1)d\)
→ \(n = \dfrac{a_n - a}{d} + 1\)
5.1. Example
AP: 5, 8, 11, 14, ...
Which term is 41?
\(41 = 5 + (n-1)3\)
\(41 - 5 = 3(n-1)\)
\(36 = 3(n-1)\)
\(n - 1 = 12\)
\(n = 13\)
Therefore, 41 is the 13th term.
6. Using nth Term to Check if a Number is in the AP
If solving for \(n\) gives a whole number, the number is part of the AP.
6.1. Example
Is 52 a term in the AP 4, 9, 14, 19, ...?
\(52 = 4 + (n-1)5\)
\(48 = 5(n-1)\)
\(n-1 = 9.6\)
n is not a whole number → 52 is not in the AP.
7. Common Mistakes
- Using n instead of (n−1) in the formula.
- Incorrect signs when the AP is decreasing.
- Mixing up 'term position' and 'term value'.
- Not checking whether n is a whole number.
8. Quick Practice
Use the nth term formula to answer:
- Find the 15th term of the AP: 2, 6, 10, 14, ...
- Find the 30th term of the AP whose a = 7 and d = -3.
- Check if 70 is a term of the AP: 1, 5, 9, 13, ...
- Which term of AP 12, 17, 22, 27, ... is 97?
9. Summary
- The nth term of an AP is given by \(a_n = a + (n - 1)d\).
- It helps find any term quickly without listing previous terms.
- You can also find term position using the rearranged formula.
- If n is not a whole number, the value is not part of the AP.