Nth Term of Arithmetic Progression (AP)
Arithmetic Progression (AP) is one of the most fundamental concepts in mathematics. It helps us identify patterns in sequences where the difference between consecutive terms remains constant. From salary increments and savings plans to mathematical modeling, AP plays an important role in everyday life.
In this article, we will explore the concept of Arithmetic Progression, understand the formula for finding the nth term, solve NCERT-based examples, and learn how AP is applied in real-world situations.
1.0What is an Arithmetic Progression (AP)?
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always constant. This constant is known as the common difference, usually denoted by d.
Examples:
- 2,5,8,11,14,… (common difference d=3)
- 15,12,9,6,3,… (common difference d=−3)
- 7,7,7,7,… (common difference d=0)
Key Terms:
- First term (a): The initial number in the sequence
- Common difference (d): The fixed amount added to each term to get the next
- Nth term (an): The term at the nth position in the sequence
2.0Real-Life Example of Arithmetic Progression
Consider a practical situation. Vishakha gets a job with a monthly salary of ₹6,000. Every year, her salary increases by ₹500. Her salary progression becomes:
Notice that every year the salary increases by ₹500. The sequence formed is:
6,000, 6,500, 7,000, 7,500, 8,000...
This is an Arithmetic Progression because the common difference is constant:
6500 - 6000 = 500
7000 - 6500 = 500
7500 - 7000 = 500
The common difference is ₹500.
3.0Finding the Nth Term of an AP
Suppose we want to find Vishakha's salary in the 15th year or even the 25th year. Adding ₹500 repeatedly would take time.
Instead, we use a formula.
Let:
- First term = a
- Common difference = d
- nth term = an
Then: an=a+(n−1)d
This is called the general term or nth term formula of an Arithmetic Progression.
4.0Derivation of the Nth Term Formula
Let's understand how the formula is formed.
Suppose the first term is a and common difference is d.
Then:
Second term:a2=a+d
Third term: a3=a+2d
Fourth term: a4=a+3d
Fifth term: a5=a+4d
Observing the pattern: an=a+(n−1)d
5.0Solved Examples
Question: Determine the complete AP whose 5th term is 19 and whose 11th term is 43.
Solution:
- Step 1: Translate the problem statements into two linear algebraic equations using the general term formula a_n = a + (n-1)d.
- For the 5th term ($a_5 = 19$):
a + (5 - 1)d = 19 \implies a + 4d = 19 \quad \text{--- (Equation 1)} - For the 11th term (a_{11} = 43):
a + (11 - 1)d = 43 \implies a + 10d = 43 \quad \text{--- (Equation 2)}
- Step 2: Solve the system of linear equations by subtracting Equation 1 from Equation 2.
(a + 10d) - (a + 4d) = 43 - 19
6d = 24
d = \frac{24}{6} = 4 - Step 3: Substitute the common difference d = 4 back into Equation 1 to find the first term (a).
a + 4(4) = 19
a + 16 = 19 \implies a = 19 - 16 = 3 - Step 4: Reconstruct the structural sequence by adding $d$ sequentially.
\text{Sequence} = a, (a+d), (a+2d), (a+3d), \dots = 3, 7, 11, 15, 19, \dots
Answer: The required Arithmetic Progression is 3, 7, 11, 15, 19, ...
Example 2: Finding a specific term value when the AP is given
Question: Find the 12th term of the AP: 5, 11, 17, 23……
Solution:
- Step 1: Identify the key operational components from the provided sequence.
- First term (a) = 5
- Common difference (d) = 11 - 5 = 6
- Target term position (n) = 12
- Step 2: Recall the general term formula.
a_n = a + (n - 1)d - Step 3: Substitute the parameters into the formula to isolate a_{12}
a_{12} = 5 + (12 - 1) \times 6
a_{12} = 5 + 11 \times 6
a_{12} = 5 + 66 = 71
Answer: The 12th term of the given AP sequence is 71.
Question: Which term of the AP: 50, 46, 42, 38…… is -10? Also, check if any specific term in this sequence equals 2.
Solution:
- Step 1: Extract the sequence parameters.
- First term (a) = 50
- Common difference (d) = 46 - 50 = -4
- Target term value (a_n) = -10
- Step 2: Use the general equation to resolve the value of n
a_n = a + (n - 1)d
-10 = 50 + (n - 1)(-4)
-10 = 50 - 4n + 4
-10 = 54 - 4n - Step 3: Isolate the variable n using basic transposition.
-4n = -10 - 54
-4n = -64
n = \frac{-64}{-4} = 16
Therefore, the 16th term of this AP is -10 - Step 4: Check if any term equals 2 by setting a_n = 2
2 = 50 + (n - 1)(-4)
2 = 54 - 4n
4n = 54 - 2
4n = 52 \implies n = \frac{52}{4} = 13
Answer: The 16th term of the sequence is $-10$, and the 13th term evaluates exactly to $2$.
6.0Applications of the Nth Term of AP
The nth term formula isn’t limited to math classes. It’s widely used in:
- Finance: Calculating future payments, savings, or investments with regular increments
- Physics: Modeling linear motion, such as constant acceleration
- Computer Science: Determining memory addresses in arrays with fixed increments
- Architecture/Construction: Planning evenly spaced structures (e.g., fence posts, windows)
- Business: Predicting sales, profits, or expenses when growth is steady
Example:
If a company’s profits increase by 1,000 every month, starting from 2,000, the profit in the 12th month is:
a12=2000+(12−1)×1000=2000+11,000=$13,000
7.0Properties of AP and Its Nth Term
- The difference between any two consecutive terms is always the common difference (d)
- If the common difference is positive, the sequence is increasing
- If the common difference is negative, the sequence is decreasing
- If the common difference is zero, the sequence is constant
- The nth term can be positive, negative, or zero depending on a, d, and n
- The sum of the first n terms (Sn) of an AP is given by:
Sn=n2[2a+(n−1)d]
8.0Common Mistakes Students Should Avoid
- Incorrect Common Difference: Always subtract consecutive terms correctly.
- Using Wrong Value of n: Remember that n indicates the term position.
- Sign Errors: Negative common differences must be handled carefully.
- Calculation Mistakes: Double-check multiplication involving ((n-1)d).