Exercise 5.2 – Question 1
Part (i)
Find: \(a_n = ?\)
📋 Given Information
- First term: \(a = 7\)
- Common difference: \(d = 3\)
- Term number: \(n = 8\)
🎯 To Find
The 8th term of the arithmetic progression: \(a_8\)
📐 Formula Used
where \(a_n\) is the \(n^{th}\) term, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
📝 Step-by-Step Solution
Let’s verify by writing the first 8 terms:
\(7, 10, 13, 16, 19, 22, 25, 28\)
The 8th term is indeed 28. ✓
Part (ii)
Find: \(d = ?\)
📋 Given Information
- First term: \(a = -18\)
- Term number: \(n = 10\)
- 10th term: \(a_{10} = 0\)
🎯 To Find
The common difference: \(d\)
📝 Step-by-Step Solution
Let’s verify: \(a_{10} = -18 + (10-1) \times 2 = -18 + 18 = 0\) ✓
The AP is: \(-18, -16, -14, -12, -10, -8, -6, -4, -2, 0\)
Part (iii)
Find: \(a = ?\)
📋 Given Information
- Common difference: \(d = -3\)
- Term number: \(n = 18\)
- 18th term: \(a_{18} = -5\)
🎯 To Find
The first term: \(a\)
📝 Step-by-Step Solution
Let’s verify: \(a_{18} = 46 + (18-1) \times (-3) = 46 – 51 = -5\) ✓
The first few terms are: \(46, 43, 40, 37, 34, \ldots\)
Part (iv)
Find: \(n = ?\)
📋 Given Information
- First term: \(a = -18.9\)
- Common difference: \(d = 2.5\)
- nth term: \(a_n = 3.6\)
🎯 To Find
The term number: \(n\)
📝 Step-by-Step Solution
Let’s verify: \(a_{10} = -18.9 + (10-1) \times 2.5 = -18.9 + 22.5 = 3.6\) ✓
📊 Summary Table
| Part | a (First Term) | d (Common Difference) | n (Term Number) | aₙ (nth Term) |
|---|---|---|---|---|
| (i) | 7 | 3 | 8 | 28 |
| (ii) | -18 | 2 | 10 | 0 |
| (iii) | 46 | -3 | 18 | -5 |
| (iv) | -18.9 | 2.5 | 10 | 3.6 |
⚠️ Common Mistakes to Avoid
❌ Mistake 1: Using \(n\) instead of \((n-1)\) in the formula.
✅ Correct: Always use \(a_n = a + (n-1)d\), not \(a_n = a + nd\).
❌ Mistake 2: Sign errors when dealing with negative values.
✅ Correct: Be careful with signs: \(-18 + 18 = 0\), not \(-18 – 18\).
❌ Mistake 3: Calculation errors with decimals.
✅ Correct: Double-check decimal arithmetic: \(22.5 ÷ 2.5 = 9\).
❌ Mistake 4: Forgetting to add 1 when finding \(n\).
✅ Correct: If \(n – 1 = 9\), then \(n = 10\), not 9.
💡 Key Points to Remember
- The nth term formula: \(a_n = a + (n-1)d\)
- To find \(a_n\): Substitute \(a\), \(d\), and \(n\) into the formula
- To find \(d\): Rearrange to \(d = \frac{a_n – a}{n – 1}\)
- To find \(a\): Rearrange to \(a = a_n – (n-1)d\)
- To find \(n\): Rearrange to \(n = \frac{a_n – a}{d} + 1\)
- Always verify your answer by substituting back into the formula
- Be careful with negative numbers and decimal calculations
