Question 2 – All Parts
Part (i)
📋 Given Information
- First term: \(a = 10\)
- Common difference: \(d = 10\)
🎯 To Find
First four terms of the arithmetic progression.
📐 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
The first term is already given:
\[ a_1 = a = 10 \]
Using the formula with \(n = 2\):
\[ \begin{aligned} a_2 &= a + (2 – 1)d \\ &= 10 + (1)(10) \\ &= 10 + 10 \\ &= 20 \end{aligned} \]Using the formula with \(n = 3\):
\[ \begin{aligned} a_3 &= a + (3 – 1)d \\ &= 10 + (2)(10) \\ &= 10 + 20 \\ &= 30 \end{aligned} \]Using the formula with \(n = 4\):
\[ \begin{aligned} a_4 &= a + (4 – 1)d \\ &= 10 + (3)(10) \\ &= 10 + 30 \\ &= 40 \end{aligned} \]Since we know the common difference is \(10\), we can simply add \(10\) to each term to get the next term:
- \(a_1 = 10\)
- \(a_2 = 10 + 10 = 20\)
- \(a_3 = 20 + 10 = 30\)
- \(a_4 = 30 + 10 = 40\)
The first four terms are: \(10, 20, 30, 40\)
Part (ii)
📋 Given Information
- First term: \(a = -2\)
- Common difference: \(d = 0\)
🎯 To Find
First four terms of the arithmetic progression.
💡 Special Case
When the common difference \(d = 0\), all terms in the AP are equal to the first term. This creates a constant sequence.
📝 Step-by-Step Solution
The first four terms are: \(-2, -2, -2, -2\) (Constant sequence)
Part (iii)
📋 Given Information
- First term: \(a = 4\)
- Common difference: \(d = -3\) (negative)
🎯 To Find
First four terms of the arithmetic progression.
💡 Special Case
When the common difference is negative, the AP is decreasing. Each term is smaller than the previous term.
📝 Step-by-Step Solution
Check that the difference between consecutive terms is constant:
\[ \begin{aligned} a_2 – a_1 &= 1 – 4 = -3 \quad ✓ \\ a_3 – a_2 &= -2 – 1 = -3 \quad ✓ \\ a_4 – a_3 &= -5 – (-2) = -3 \quad ✓ \end{aligned} \]The first four terms are: \(4, 1, -2, -5\)
Part (iv)
📋 Given Information
- First term: \(a = -1\)
- Common difference: \(d = \frac{1}{2}\) (fractional)
🎯 To Find
First four terms of the arithmetic progression.
📝 Step-by-Step Solution
The terms can also be written in decimal form:
- \(a_1 = -1 = -1.0\)
- \(a_2 = -\frac{1}{2} = -0.5\)
- \(a_3 = 0 = 0.0\)
- \(a_4 = \frac{1}{2} = 0.5\)
The first four terms are: \(-1, -\frac{1}{2}, 0, \frac{1}{2}\)
Part (v)
📋 Given Information
- First term: \(a = -1.25\)
- Common difference: \(d = -0.25\) (negative decimal)
🎯 To Find
First four terms of the arithmetic progression.
📝 Step-by-Step Solution
The decimal values can also be expressed as fractions:
- \(a_1 = -1.25 = -\frac{5}{4}\)
- \(a_2 = -1.50 = -\frac{3}{2}\)
- \(a_3 = -1.75 = -\frac{7}{4}\)
- \(a_4 = -2.00 = -2\)
Common difference: \(d = -0.25 = -\frac{1}{4}\)
The first four terms are: \(-1.25, -1.50, -1.75, -2.00\)
⚠️ Common Mistakes to Avoid
❌ Mistake 1: Forgetting to multiply \(d\) by \((n-1)\) 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 common difference.
✅ Correct: When \(d\) is negative, remember that adding a negative number means subtraction: \(a + (-3) = a – 3\).
❌ Mistake 3: Calculation errors with fractions and decimals.
✅ Correct: Be careful when adding/subtracting fractions. Convert to common denominators or use decimal equivalents consistently.
❌ Mistake 4: Thinking that when \(d = 0\), there is no AP.
✅ Correct: When \(d = 0\), it’s still an AP—just a constant sequence where all terms are equal.
💡 Key Points to Remember
- The nth term formula for an AP is: \(a_n = a + (n-1)d\)
- When \(d > 0\): AP is increasing (each term is larger than the previous)
- When \(d < 0\): AP is decreasing (each term is smaller than the previous)
- When \(d = 0\): AP is a constant sequence (all terms are equal)
- The common difference can be positive, negative, zero, fractional, or decimal
- To find any term, you only need the first term \(a\) and common difference \(d\)
- Quick method: Add \(d\) repeatedly to get successive terms: \(a, a+d, a+2d, a+3d, \ldots\)
🎯 Practice Problems
Try these problems to test your understanding:
