📚 Question 2(b)
Write first four terms of the AP, when the first term \(a\) and the common difference \(d\) are given as follows:
\(a = -2\), \(d = 0\)
📌 Given Information
- First term: \(a = -2\)
- Common difference: \(d = 0\)
- We need to find the first four terms of the AP
🎯 To Find
The first four terms of the Arithmetic Progression: \(a_1, a_2, a_3, a_4\)
💡 Key Concept
Special Case: When \(d = 0\)
When the common difference is zero, all terms in the AP are equal. This creates a constant sequence.
\[a_n = a + (n-1) \times 0 = a\]
Important: A constant sequence is still considered an AP with \(d = 0\).
✍️ Step-by-Step Solution
Step 1: Identify the given values
First term: \(a = -2\)
Common difference: \(d = 0\)
Step 2: Calculate each term using the formula
Using the formula \(a_n = a + (n-1)d\):
First term (\(n = 1\)):
\(a_1 = a = -2\)
Second term (\(n = 2\)):
\(a_2 = a + (2-1)d\)
\(a_2 = -2 + (1)(0)\)
\(a_2 = -2 + 0 = -2\)
Third term (\(n = 3\)):
\(a_3 = a + (3-1)d\)
\(a_3 = -2 + (2)(0)\)
\(a_3 = -2 + 0 = -2\)
Fourth term (\(n = 4\)):
\(a_4 = a + (4-1)d\)
\(a_4 = -2 + (3)(0)\)
\(a_4 = -2 + 0 = -2\)
Step 3: Verify the common difference
Check that the difference between consecutive terms is constant:
\(a_2 – a_1 = -2 – (-2) = 0\) ✓
\(a_3 – a_2 = -2 – (-2) = 0\) ✓
\(a_4 – a_3 = -2 – (-2) = 0\) ✓
Verification: All differences equal 0, confirming this is a valid AP (constant sequence).
✅ Final Answer
The first four terms of the AP are:
\[-2, -2, -2, -2\]
📊 Visual Representation
Pattern: All terms remain constant at -2
🔍 Understanding Constant Sequences
What is a Constant Sequence?
A constant sequence is a special type of AP where:
- The common difference \(d = 0\)
- All terms are identical
- No growth or decline occurs
- It’s neither increasing nor decreasing
Real-life example: A fixed monthly salary (without increments) forms a constant sequence: ₹30,000, ₹30,000, ₹30,000, …
🔍 Properties of This AP
⚠️ Common Mistakes to Avoid
- Thinking it’s not an AP: A constant sequence IS an AP with \(d = 0\).
- Confusing with zero sequence: This is -2, -2, -2, … not 0, 0, 0, …
- Sign errors: \(-2 – (-2) = 0\), not \(-4\). Be careful with negative numbers.
- Assuming terms must change: In an AP, terms can remain the same if \(d = 0\).
📝 Practice Problems
- Write the first 10 terms of the AP with \(a = -2\) and \(d = 0\).
- Find the 50th term of this AP.
- If \(a = 5\) and \(d = 0\), what is the sum of the first 20 terms?
- Can you have an AP where \(a = 0\) and \(d = 0\)? What would it look like?
🔗 Related Topics
Written by Farhan Mansuri
M.Sc. Mathematics | B.Ed. | 15+ Years Teaching Experience
Farhan Mansuri is a dedicated mathematics educator with over 15 years of experience teaching CBSE curriculum. He specializes in making complex mathematical concepts accessible to Class 10 students.
🔗 Connect on LinkedIn