Question 4 – All Parts

Which of the following are APs? If they form an AP, find the common difference \(d\) and write three more terms.

Part (i)

\(2, 4, 8, 16, \ldots\)

📋 Given Sequence

\(2, 4, 8, 16, \ldots\)

🎯 To Determine

Whether this sequence forms an Arithmetic Progression or not.

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= 4 – 2 = 2 \\ a_3 – a_2 &= 8 – 4 = 4 \\ a_4 – a_3 &= 16 – 8 = 8 \end{aligned} \]

2 Check if differences are constant:

The differences are: \(2, 4, 8, \ldots\)

These differences are NOT constant.

3 Conclusion:

Since the difference between consecutive terms is not constant, this sequence is NOT an Arithmetic Progression.

Note: This is a Geometric Progression (GP) with common ratio \(r = 2\).

❌ Answer: NOT an AP (It’s a GP with ratio 2)

Part (ii)

\(2, \frac{5}{2}, 3, \frac{7}{2}, \ldots\)

📋 Given Sequence

\(2, \frac{5}{2}, 3, \frac{7}{2}, \ldots\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= \frac{5}{2} – 2 = \frac{5}{2} – \frac{4}{2} = \frac{1}{2} \\ a_3 – a_2 &= 3 – \frac{5}{2} = \frac{6}{2} – \frac{5}{2} = \frac{1}{2} \\ a_4 – a_3 &= \frac{7}{2} – 3 = \frac{7}{2} – \frac{6}{2} = \frac{1}{2} \end{aligned} \]

2 Check if differences are constant:

All differences equal \(\frac{1}{2}\), which is constant.

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = \frac{1}{2}\)

3 Find next three terms:

Add \(d = \frac{1}{2}\) to each previous term:

\[ \begin{aligned} a_5 &= \frac{7}{2} + \frac{1}{2} = \frac{8}{2} = 4 \\ a_6 &= 4 + \frac{1}{2} = \frac{9}{2} \\ a_7 &= \frac{9}{2} + \frac{1}{2} = \frac{10}{2} = 5 \end{aligned} \]

✅ Answer: Yes, it’s an AP with \(d = \frac{1}{2}\)
Next three terms: \(4, \frac{9}{2}, 5\)

Part (iii)

\(-1.2, -3.2, -5.2, -7.2, \ldots\)

📋 Given Sequence

\(-1.2, -3.2, -5.2, -7.2, \ldots\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= -3.2 – (-1.2) = -3.2 + 1.2 = -2 \\ a_3 – a_2 &= -5.2 – (-3.2) = -5.2 + 3.2 = -2 \\ a_4 – a_3 &= -7.2 – (-5.2) = -7.2 + 5.2 = -2 \end{aligned} \]

2 Check if differences are constant:

All differences equal \(-2\), which is constant.

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = -2\)

3 Find next three terms:

Add \(d = -2\) to each previous term:

\[ \begin{aligned} a_5 &= -7.2 + (-2) = -7.2 – 2 = -9.2 \\ a_6 &= -9.2 + (-2) = -9.2 – 2 = -11.2 \\ a_7 &= -11.2 + (-2) = -11.2 – 2 = -13.2 \end{aligned} \]

✅ Answer: Yes, it’s an AP with \(d = -2\)
Next three terms: \(-9.2, -11.2, -13.2\)

Part (iv)

\(-10, -6, -2, 2, \ldots\)

📋 Given Sequence

\(-10, -6, -2, 2, \ldots\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= -6 – (-10) = -6 + 10 = 4 \\ a_3 – a_2 &= -2 – (-6) = -2 + 6 = 4 \\ a_4 – a_3 &= 2 – (-2) = 2 + 2 = 4 \end{aligned} \]

2 Check if differences are constant:

All differences equal \(4\), which is constant.

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = 4\)

3 Find next three terms:

Add \(d = 4\) to each previous term:

\[ \begin{aligned} a_5 &= 2 + 4 = 6 \\ a_6 &= 6 + 4 = 10 \\ a_7 &= 10 + 4 = 14 \end{aligned} \]

✅ Answer: Yes, it’s an AP with \(d = 4\)
Next three terms: \(6, 10, 14\)

Part (v)

\(3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}, \ldots\)

📋 Given Sequence

\(3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}, \ldots\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= (3 + \sqrt{2}) – 3 = \sqrt{2} \\ a_3 – a_2 &= (3 + 2\sqrt{2}) – (3 + \sqrt{2}) = \sqrt{2} \\ a_4 – a_3 &= (3 + 3\sqrt{2}) – (3 + 2\sqrt{2}) = \sqrt{2} \end{aligned} \]

2 Check if differences are constant:

All differences equal \(\sqrt{2}\), which is constant.

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = \sqrt{2}\)

3 Find next three terms:

Add \(d = \sqrt{2}\) to each previous term:

\[ \begin{aligned} a_5 &= 3 + 3\sqrt{2} + \sqrt{2} = 3 + 4\sqrt{2} \\ a_6 &= 3 + 4\sqrt{2} + \sqrt{2} = 3 + 5\sqrt{2} \\ a_7 &= 3 + 5\sqrt{2} + \sqrt{2} = 3 + 6\sqrt{2} \end{aligned} \]

✅ Answer: Yes, it’s an AP with \(d = \sqrt{2}\)
Next three terms: \(3 + 4\sqrt{2}, 3 + 5\sqrt{2}, 3 + 6\sqrt{2}\)

Part (vi)

\(0.2, 0.22, 0.222, 0.2222, \ldots\)

📋 Given Sequence

\(0.2, 0.22, 0.222, 0.2222, \ldots\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= 0.22 – 0.2 = 0.02 \\ a_3 – a_2 &= 0.222 – 0.22 = 0.002 \\ a_4 – a_3 &= 0.2222 – 0.222 = 0.0002 \end{aligned} \]

2 Check if differences are constant:

The differences are: \(0.02, 0.002, 0.0002, \ldots\)

These differences are NOT constant.

3 Conclusion:

Since the difference between consecutive terms is not constant, this sequence is NOT an Arithmetic Progression.

❌ Answer: NOT an AP

Part (vii)

\(0, -4, -8, -12, \ldots\)

📋 Given Sequence

\(0, -4, -8, -12, \ldots\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= -4 – 0 = -4 \\ a_3 – a_2 &= -8 – (-4) = -8 + 4 = -4 \\ a_4 – a_3 &= -12 – (-8) = -12 + 8 = -4 \end{aligned} \]

2 Check if differences are constant:

All differences equal \(-4\), which is constant.

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = -4\)

3 Find next three terms:

Add \(d = -4\) to each previous term:

\[ \begin{aligned} a_5 &= -12 + (-4) = -16 \\ a_6 &= -16 + (-4) = -20 \\ a_7 &= -20 + (-4) = -24 \end{aligned} \]

✅ Answer: Yes, it’s an AP with \(d = -4\)
Next three terms: \(-16, -20, -24\)

Part (viii)

\(-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, \ldots\)

📋 Given Sequence

\(-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, \ldots\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= -\frac{1}{2} – \left(-\frac{1}{2}\right) = 0 \\ a_3 – a_2 &= -\frac{1}{2} – \left(-\frac{1}{2}\right) = 0 \\ a_4 – a_3 &= -\frac{1}{2} – \left(-\frac{1}{2}\right) = 0 \end{aligned} \]

2 Check if differences are constant:

All differences equal \(0\), which is constant.

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = 0\)

Note: This is a constant sequence (all terms are equal).

3 Find next three terms:

Since \(d = 0\), all terms remain the same:

\[ a_5 = a_6 = a_7 = -\frac{1}{2} \]

✅ Answer: Yes, it’s an AP with \(d = 0\)
Next three terms: \(-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}\)

Part (ix)

\(1, 3, 9, 27, \ldots\)

📋 Given Sequence

\(1, 3, 9, 27, \ldots\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= 3 – 1 = 2 \\ a_3 – a_2 &= 9 – 3 = 6 \\ a_4 – a_3 &= 27 – 9 = 18 \end{aligned} \]

2 Check if differences are constant:

The differences are: \(2, 6, 18, \ldots\)

These differences are NOT constant.

3 Conclusion:

Since the difference between consecutive terms is not constant, this sequence is NOT an Arithmetic Progression.

Note: This is a Geometric Progression (GP) with common ratio \(r = 3\).

❌ Answer: NOT an AP (It’s a GP with ratio 3)

Part (x)

\(a, 2a, 3a, 4a, \ldots\) where \(a \neq 0\)

📋 Given Sequence

\(a, 2a, 3a, 4a, \ldots\) where \(a \neq 0\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= 2a – a = a \\ a_3 – a_2 &= 3a – 2a = a \\ a_4 – a_3 &= 4a – 3a = a \end{aligned} \]

2 Check if differences are constant:

All differences equal \(a\), which is constant (since \(a\) is a fixed value).

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = a\)

3 Find next three terms:

Add \(d = a\) to each previous term:

\[ \begin{aligned} a_5 &= 4a + a = 5a \\ a_6 &= 5a + a = 6a \\ a_7 &= 6a + a = 7a \end{aligned} \]

✅ Answer: Yes, it’s an AP with \(d = a\)
Next three terms: \(5a, 6a, 7a\)

Part (xi)

\(a, a^2, a^3, a^4, \ldots\) where \(a \neq 1\)

📋 Given Sequence

\(a, a^2, a^3, a^4, \ldots\) where \(a \neq 1\)

📝 Solution

1 Calculate differences between consecutive terms: \[ \begin{aligned} a_2 – a_1 &= a^2 – a = a(a – 1) \\ a_3 – a_2 &= a^3 – a^2 = a^2(a – 1) \\ a_4 – a_3 &= a^4 – a^3 = a^3(a – 1) \end{aligned} \]

2 Check if differences are constant:

The differences are: \(a(a-1), a^2(a-1), a^3(a-1), \ldots\)

These are NOT constant (unless \(a = 1\), which is excluded).

3 Conclusion:

Since the difference between consecutive terms is not constant, this sequence is NOT an Arithmetic Progression.

Note: This is a Geometric Progression (GP) with common ratio \(r = a\).

❌ Answer: NOT an AP (It’s a GP with ratio a)

Part (xii)

\(\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots\)

📋 Given Sequence

\(\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots\)

📝 Solution

1 Simplify the terms: \[ \begin{aligned} a_1 &= \sqrt{2} = \sqrt{2} \\ a_2 &= \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \\ a_3 &= \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \\ a_4 &= \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \end{aligned} \]

So the sequence becomes: \(\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \ldots\)

2 Calculate differences: \[ \begin{aligned} a_2 – a_1 &= 2\sqrt{2} – \sqrt{2} = \sqrt{2} \\ a_3 – a_2 &= 3\sqrt{2} – 2\sqrt{2} = \sqrt{2} \\ a_4 – a_3 &= 4\sqrt{2} – 3\sqrt{2} = \sqrt{2} \end{aligned} \]

3 Check if differences are constant:

All differences equal \(\sqrt{2}\), which is constant.

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = \sqrt{2}\)

4 Find next three terms: \[ \begin{aligned} a_5 &= 4\sqrt{2} + \sqrt{2} = 5\sqrt{2} = \sqrt{50} \\ a_6 &= 5\sqrt{2} + \sqrt{2} = 6\sqrt{2} = \sqrt{72} \\ a_7 &= 6\sqrt{2} + \sqrt{2} = 7\sqrt{2} = \sqrt{98} \end{aligned} \]

✅ Answer: Yes, it’s an AP with \(d = \sqrt{2}\)
Next three terms: \(\sqrt{50}, \sqrt{72}, \sqrt{98}\) or \(5\sqrt{2}, 6\sqrt{2}, 7\sqrt{2}\)

Part (xiii)

\(\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \ldots\)

📋 Given Sequence

\(\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \ldots\)

📝 Solution

1 Simplify the terms: \[ \begin{aligned} a_1 &= \sqrt{3} \\ a_2 &= \sqrt{6} \\ a_3 &= \sqrt{9} = 3 \\ a_4 &= \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \end{aligned} \]

2 Calculate differences: \[ \begin{aligned} a_2 – a_1 &= \sqrt{6} – \sqrt{3} \approx 2.449 – 1.732 = 0.717 \\ a_3 – a_2 &= 3 – \sqrt{6} \approx 3 – 2.449 = 0.551 \\ a_4 – a_3 &= 2\sqrt{3} – 3 \approx 3.464 – 3 = 0.464 \end{aligned} \]

3 Check if differences are constant:

The differences are approximately: \(0.717, 0.551, 0.464, \ldots\)

These differences are NOT constant.

4 Conclusion:

Since the difference between consecutive terms is not constant, this sequence is NOT an Arithmetic Progression.

❌ Answer: NOT an AP

Part (xiv)

\(1^2, 3^2, 5^2, 7^2, \ldots\)

📋 Given Sequence

\(1^2, 3^2, 5^2, 7^2, \ldots\)

📝 Solution

1 Calculate the values: \[ \begin{aligned} a_1 &= 1^2 = 1 \\ a_2 &= 3^2 = 9 \\ a_3 &= 5^2 = 25 \\ a_4 &= 7^2 = 49 \end{aligned} \]

So the sequence is: \(1, 9, 25, 49, \ldots\)

2 Calculate differences: \[ \begin{aligned} a_2 – a_1 &= 9 – 1 = 8 \\ a_3 – a_2 &= 25 – 9 = 16 \\ a_4 – a_3 &= 49 – 25 = 24 \end{aligned} \]

3 Check if differences are constant:

The differences are: \(8, 16, 24, \ldots\)

These differences are NOT constant.

4 Conclusion:

Since the difference between consecutive terms is not constant, this sequence is NOT an Arithmetic Progression.

Note: This is the sequence of squares of odd numbers.

❌ Answer: NOT an AP

Part (xv)

\(1^2, 5^2, 7^2, 73, \ldots\)

📋 Given Sequence

\(1^2, 5^2, 7^2, 73, \ldots\)

📝 Solution

1 Calculate the values: \[ \begin{aligned} a_1 &= 1^2 = 1 \\ a_2 &= 5^2 = 25 \\ a_3 &= 7^2 = 49 \\ a_4 &= 73 \end{aligned} \]

So the sequence is: \(1, 25, 49, 73, \ldots\)

2 Calculate differences: \[ \begin{aligned} a_2 – a_1 &= 25 – 1 = 24 \\ a_3 – a_2 &= 49 – 25 = 24 \\ a_4 – a_3 &= 73 – 49 = 24 \end{aligned} \]

3 Check if differences are constant:

All differences equal \(24\), which is constant.

Therefore, this IS an Arithmetic Progression.

Common difference: \(d = 24\)

4 Find next three terms: \[ \begin{aligned} a_5 &= 73 + 24 = 97 \\ a_6 &= 97 + 24 = 121 \\ a_7 &= 121 + 24 = 145 \end{aligned} \]

✅ Answer: Yes, it’s an AP with \(d = 24\)
Next three terms: \(97, 121, 145\)

📊 Summary Table

Part	Sequence	Is it AP?	Common Difference (d)	Next Three Terms
(i)	2, 4, 8, 16, …	❌ No	—	GP with ratio 2
(ii)	2, 5/2, 3, 7/2, …	✅ Yes	1/2	4, 9/2, 5
(iii)	-1.2, -3.2, -5.2, -7.2, …	✅ Yes	-2	-9.2, -11.2, -13.2
(iv)	-10, -6, -2, 2, …	✅ Yes	4	6, 10, 14
(v)	3, 3+√2, 3+2√2, 3+3√2, …	✅ Yes	√2	3+4√2, 3+5√2, 3+6√2
(vi)	0.2, 0.22, 0.222, 0.2222, …	❌ No	—	Not an AP
(vii)	0, -4, -8, -12, …	✅ Yes	-4	-16, -20, -24
(viii)	-1/2, -1/2, -1/2, -1/2, …	✅ Yes	0	-1/2, -1/2, -1/2
(ix)	1, 3, 9, 27, …	❌ No	—	GP with ratio 3
(x)	a, 2a, 3a, 4a, …	✅ Yes	a	5a, 6a, 7a
(xi)	a, a², a³, a⁴, …	❌ No	—	GP with ratio a
(xii)	√2, √8, √18, √32, …	✅ Yes	√2	√50, √72, √98
(xiii)	√3, √6, √9, √12, …	❌ No	—	Not an AP
(xiv)	1², 3², 5², 7², …	❌ No	—	Not an AP
(xv)	1², 5², 7², 73, …	✅ Yes	24	97, 121, 145

⚠️ Common Mistakes to Avoid

❌ Mistake 1: Assuming all sequences are APs without checking differences.

✅ Correct: Always calculate at least 2-3 consecutive differences to verify if they’re constant.

❌ Mistake 2: Not simplifying radical terms before checking for AP.

✅ Correct: Simplify terms like √8 = 2√2 before calculating differences.

❌ Mistake 3: Confusing geometric progressions with arithmetic progressions.

✅ Correct: GP has constant ratio (multiply), AP has constant difference (add/subtract).

❌ Mistake 4: Calculation errors with negative numbers.

✅ Correct: Remember: subtracting a negative is addition. Example: -6 – (-10) = -6 + 10 = 4

❌ Mistake 5: Thinking d = 0 means it’s not an AP.

✅ Correct: When d = 0, it’s still an AP (constant sequence).

💡 Key Points to Remember

An Arithmetic Progression (AP) has a constant difference between consecutive terms
Formula to check: \(d = a_{n+1} – a_n\) must be constant for all consecutive terms
Common difference can be positive, negative, zero, fractional, or irrational
When \(d > 0\): AP is increasing
When \(d < 0\): AP is decreasing
When \(d = 0\): AP is a constant sequence
Always simplify radical and fractional terms before checking differences
Geometric Progressions (GP) have constant ratios, not constant differences
To find next terms: add common difference \(d\) to the last known term
Verify your answer by checking multiple consecutive differences

Question 4 – All Parts

📑 Question Parts:

Part (i)

📋 Given Sequence

🎯 To Determine

📝 Solution

Part (ii)

📋 Given Sequence

📝 Solution

Part (iii)

📋 Given Sequence

📝 Solution

Part (iv)

📋 Given Sequence

📝 Solution

Part (v)

📋 Given Sequence

📝 Solution

Part (vi)

📋 Given Sequence

📝 Solution

Part (vii)

📋 Given Sequence

📝 Solution

Part (viii)

📋 Given Sequence

📝 Solution

Part (ix)

📋 Given Sequence

📝 Solution

Part (x)

📋 Given Sequence

📝 Solution

Part (xi)

📋 Given Sequence

📝 Solution

Part (xii)

📋 Given Sequence

📝 Solution

Part (xiii)

📋 Given Sequence

📝 Solution

Part (xiv)

📋 Given Sequence

📝 Solution

Part (xv)

📋 Given Sequence

📝 Solution

📊 Summary Table

⚠️ Common Mistakes to Avoid

💡 Key Points to Remember

📚 Related Questions

Farhan Mansuri

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