Understanding the Nth Term of an Arithmetic Progression (AP)
An Arithmetic Progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference, typically denoted by ‘d’. The formula for the nth term of an AP is a fundamental concept in algebra, allowing us to find any term in the sequence without listing all the preceding terms.
Key Concepts of an AP
- First Term (a): The starting number of the arithmetic progression.
- Common Difference (d): The constant value added to any term to get the next term. It can be positive, negative, or zero.
- Number of Terms (n): The position of the term in the sequence (e.g., for the 5th term, n=5).
- Nth Term (a_n): The term at the nth position in the sequence.
Derivation of the Nth Term Formula: a_n = a + (n-1)d
The formula for the nth term of an AP can be derived logically by observing the pattern of the terms:
- First term (a1): This is simply ‘a’. So,
a1 = a - Second term (a2): To get the second term, we add the common difference ‘d’ to the first term. So,
a2 = a + d - Third term (a3): To get the third term, we add ‘d’ to the second term. So,
a3 = a2 + d = (a + d) + d = a + 2d - Fourth term (a4): Similarly,
a4 = a3 + d = (a + 2d) + d = a + 3d
Observing this pattern, we can see that the coefficient of ‘d’ is always one less than the term number.
- For the 1st term, ‘d’ is added 0 times (1-1 = 0).
- For the 2nd term, ‘d’ is added 1 time (2-1 = 1).
- For the 3rd term, ‘d’ is added 2 times (3-1 = 2).
- For the 4th term, ‘d’ is added 3 times (4-1 = 3).
(n-1) times to the first term ‘a’.
This leads to the general formula for the nth term of an AP:
a_n = a + (n-1)d
Summary Table of Terms
| Term Number | Term in AP | Formula |
|---|---|---|
| 1st | a | a + (1-1)d = a |
| 2nd | a, a+d | a + (2-1)d = a + d |
| 3rd | a, a+d, a+2d | a + (3-1)d = a + 2d |
| nth | … | a + (n-1)d |
Quick Revision Points
- An AP is a sequence with a constant common difference.
- The first term is ‘a’.
- The common difference is ‘d’ (
a_k - a_(k-1)). - The formula for the nth term is
a_n = a + (n-1)d. - This formula is derived by recognizing that ‘d’ is added
(n-1)times to the first term to reach the nth term.
Practice Questions
Try solving these additional problems to solidify your understanding:
- Find the 10th term of an AP: 2, 7, 12, …
- An AP has its first term as 5 and common difference as 3. What is its 15th term?
- If the 3rd term of an AP is 7 and the 7th term is 19, find the common difference and the first term.
- Which term of the AP: 21, 18, 15, … is -81?
- Derive the formula for the nth term of an AP starting from its definition and explaining each step.
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