📚 Question 2(a)
Write first four terms of the AP, when the first term \(a\) and the common difference \(d\) are given as follows:
\(a = 10\), \(d = 10\)
📌 Given Information
- First term: \(a = 10\)
- Common difference: \(d = 10\)
- 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
Arithmetic Progression (AP): A sequence where each term is obtained by adding a fixed number (common difference) to the previous term.
General Form:
\[a, a+d, a+2d, a+3d, \ldots\]
nth Term Formula:
\[a_n = a + (n-1)d\]
✍️ Step-by-Step Solution
Step 1: Identify the given values
First term: \(a = 10\)
Common difference: \(d = 10\)
Step 2: Calculate each term using the formula
Using the formula \(a_n = a + (n-1)d\):
First term (\(n = 1\)):
\(a_1 = a = 10\)
Second term (\(n = 2\)):
\(a_2 = a + (2-1)d\)
\(a_2 = 10 + (1)(10)\)
\(a_2 = 10 + 10 = 20\)
Third term (\(n = 3\)):
\(a_3 = a + (3-1)d\)
\(a_3 = 10 + (2)(10)\)
\(a_3 = 10 + 20 = 30\)
Fourth term (\(n = 4\)):
\(a_4 = a + (4-1)d\)
\(a_4 = 10 + (3)(10)\)
\(a_4 = 10 + 30 = 40\)
Step 3: Verify the common difference
Check that the difference between consecutive terms is constant:
\(a_2 – a_1 = 20 – 10 = 10\) ✓
\(a_3 – a_2 = 30 – 20 = 10\) ✓
\(a_4 – a_3 = 40 – 30 = 10\) ✓
Verification: All differences equal 10, confirming this is a valid AP.
✅ Final Answer
The first four terms of the AP are:
\[10, 20, 30, 40\]
📊 Visual Representation
Pattern: Each term increases by 10
🔄 Alternative Method: Direct Addition
Instead of using the formula, we can simply add the common difference repeatedly:
Start with \(a_1 = 10\)
\(a_2 = a_1 + d = 10 + 10 = 20\)
\(a_3 = a_2 + d = 20 + 10 = 30\)
\(a_4 = a_3 + d = 30 + 10 = 40\)
Note: Both methods give the same result. Use whichever is more convenient!
🔍 Properties of This AP
⚠️ Common Mistakes to Avoid
- Starting from \(n = 0\): The first term corresponds to \(n = 1\), not \(n = 0\).
- Forgetting to multiply by \((n-1)\): The formula is \(a + (n-1)d\), not \(a + nd\).
- Confusing \(a\) and \(d\): Make sure to identify which is the first term and which is the common difference.
- Not verifying: Always check that consecutive differences are equal to confirm it’s an AP.
📝 Practice Problems
- Find the first 6 terms of the AP with \(a = 10\) and \(d = 10\).
- What is the 20th term of this AP?
- If the first term is 10 and the 5th term is 50, find the common difference.
- Find the sum of the first four terms: \(10 + 20 + 30 + 40 = ?\)
🔗 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.
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