Exercise 5.1, Question 1
In which situation does the list of numbers form an AP? Give a reason.
(i) A taxi charges ₹15 for the first km and ₹8 for each extra km; the list is the fare after 1 km, 2 km, 3 km, …
Show solution
- Fares: \(15, 23, 31, 39, \ldots\)
- Each fare is ₹8 more than the one before, so the difference is constant.
Answer: Yes, an AP with \(a = 15,\ d = 8\).
(ii) A pump removes \(\tfrac14\) of the air left in a cylinder each time; the list is the amount of air left after each stroke.
Show solution
- Start with amount \(x\). After each stroke \(\tfrac34\) of what was there remains: \(x, \tfrac34x, \tfrac{9}{16}x, \ldots\)
- Differences: \(-\tfrac14x\), then \(-\tfrac{3}{16}x\). They are not equal.
Answer: No: each amount is \(\tfrac34\) of the last (it is multiplied, not added to).
(iii) Digging a well costs ₹150 for the first metre and ₹50 more for each metre after that; the list is the cost after 1 m, 2 m, 3 m, …
Show solution
- Costs: \(150, 200, 250, 300, \ldots\)
- Each cost is ₹50 more than the previous one.
Answer: Yes, an AP with \(a = 150,\ d = 50\).
(iv) ₹10000 is invested at 8% compound interest a year; the list is the amount at the end of each year.
Show solution
- Amounts: \[\begin{gathered}10000(1.08), \\ 10000(1.08)^2, \\ 10000(1.08)^3, \ldots\end{gathered}\] that is \(10800, 11664, 12597.12, \ldots\)
- Differences: \(864\), then \(933.12\). Not equal.
Answer: No: compound interest multiplies by 1.08 each year.
Practise this: Step 1, Secure the basics →
Exercise 5.1, Question 2
Write the first four terms of the AP.
(i) \(a = 10,\ d = 10\)
Show solution
- Add \(10\) each time.
Answer: \(10, 20, 30, 40\)
(ii) \(a = -2,\ d = 0\)
Show solution
- With \(d = 0\) every term equals the first.
Answer: \(-2, -2, -2, -2\)
(iii) \(a = 4,\ d = -3\)
Show solution
- Subtract \(3\) each time.
Answer: \(4, 1, -2, -5\)
(iv) \(a = -1,\ d = \tfrac12\)
Show solution
- Add \(\tfrac12\) each time.
Answer: \(-1, -\tfrac12, 0, \tfrac12\)
(v) \(a = -1.25,\ d = -0.25\)
Show solution
- Subtract \(0.25\) each time.
Answer: \(-1.25, -1.50, -1.75, -2.00\)
Practise this: Step 1, Secure the basics →
Exercise 5.1, Question 3
Write the first term and the common difference.
(i) \(3, 1, -1, -3, \ldots\)
Show solution
- \(a = 3\).
- \(d = 1 - 3 = -2\) (and \(-1 - 1 = -2\) too).
Answer: \(a = 3,\ d = -2\)
(ii) \(-5, -1, 3, 7, \ldots\)
Show solution
- \(a = -5\).
- \(d = -1 - (-5) = 4\).
Answer: \(a = -5,\ d = 4\)
(iii) \(\tfrac13, \tfrac53, \tfrac93, \tfrac{13}{3}, \ldots\)
Show solution
- \(a = \tfrac13\).
- \(d = \tfrac53 - \tfrac13 = \tfrac43\).
Answer: \(a = \tfrac13,\ d = \tfrac43\)
(iv) \(0.6, 1.7, 2.8, 3.9, \ldots\)
Show solution
- \(a = 0.6\).
- \(d = 1.7 - 0.6 = 1.1\).
Answer: \(a = 0.6,\ d = 1.1\)
Practise this: Step 1, Secure the basics →
Exercise 5.1, Question 4
Which of these are APs? For each AP, give \(d\) and the next three terms.
(i) \(2, 4, 8, 16, \ldots\)
Show solution
- Differences: \(2, 4, 8\), not constant.
Answer: Not an AP
(ii) \(2, \tfrac52, 3, \tfrac72, \ldots\)
Show solution
- Differences are all \(\tfrac12\).
Answer: AP, \(d = \tfrac12\); next: \(4, \tfrac92, 5\)
(iii) \(-1.2, -3.2, -5.2, -7.2, \ldots\)
Show solution
- Differences are all \(-2\).
Answer: AP, \(d = -2\); next: \(-9.2, -11.2, -13.2\)
(iv) \(-10, -6, -2, 2, \ldots\)
Show solution
- Differences are all \(4\).
Answer: AP, \(d = 4\); next: \(6, 10, 14\)
(v) \(3, 3 + \sqrt2, 3 + 2\sqrt2, 3 + 3\sqrt2, \ldots\)
Show solution
- Differences are all \(\sqrt2\).
Answer: AP, \(d = \sqrt2\); next: \(3 + 4\sqrt2,\ 3 + 5\sqrt2,\ 3 + 6\sqrt2\)
(vi) \(0.2, 0.22, 0.222, 0.2222, \ldots\)
Show solution
- Differences: \(0.02, 0.002, 0.0002\), not constant.
Answer: Not an AP
(vii) \(0, -4, -8, -12, \ldots\)
Show solution
- Differences are all \(-4\).
Answer: AP, \(d = -4\); next: \(-16, -20, -24\)
(viii) \(-\tfrac12, -\tfrac12, -\tfrac12, -\tfrac12, \ldots\)
Show solution
- Differences are all \(0\): a constant list is an AP with \(d = 0\).
Answer: AP, \(d = 0\); next: \(-\tfrac12, -\tfrac12, -\tfrac12\)
(ix) \(1, 3, 9, 27, \ldots\)
Show solution
- Differences: \(2, 6, 18\), not constant.
Answer: Not an AP
(x) \(a, 2a, 3a, 4a, \ldots\)
Show solution
- Differences are all \(a\).
Answer: AP, \(d = a\); next: \(5a, 6a, 7a\)
(xi) \(a, a^2, a^3, a^4, \ldots\)
Show solution
- Differences: \(a^2 - a = a(a-1)\), then \(a^3 - a^2 = a^2(a-1)\).
- These are equal only when \(a = 0\) or \(a = 1\), so in general the list is not an AP.
Answer: Not an AP (in general)
(xii) \(\sqrt2, \sqrt8, \sqrt{18}, \sqrt{32}, \ldots\)
Show solution
- Simplify: \(\sqrt2, 2\sqrt2, 3\sqrt2, 4\sqrt2\).
- Differences are all \(\sqrt2\).
Answer: AP, \(d = \sqrt2\); next: \(\sqrt{50}, \sqrt{72}, \sqrt{98}\) (that is \(5\sqrt2, 6\sqrt2, 7\sqrt2\))
(xiii) \(\sqrt3, \sqrt6, \sqrt9, \sqrt{12}, \ldots\)
Show solution
- \(\sqrt6 - \sqrt3 = \sqrt3(\sqrt2 - 1)\), but \(\sqrt9 - \sqrt6 = 3 - \sqrt6\), which is different.
Answer: Not an AP
(xiv) \(1^2, 3^2, 5^2, 7^2, \ldots\)
Show solution
- Terms \(1, 9, 25, 49\); differences \(8, 16, 24\), not constant.
Answer: Not an AP
(xv) \(1^2, 5^2, 7^2, 73, \ldots\)
Show solution
- Terms \(1, 25, 49, 73\); differences are all \(24\).
Answer: AP, \(d = 24\); next: \(97, 121, 145\)
Practise this: Step 1, Secure the basics →