Arithmetic Progressions (AP)

Q1. In which of the following situations does the list of numbers form an AP?

(i) Taxi fare for first km is ₹15, and ₹8 for each subsequent km.

$Fares: 15, 23, 31, 39, \dots$

Common difference:

$d = 23 - 15 = 8$

✅ Yes, it is an AP with $a = 15, d = 8$ .

(ii) Air is pumped out of a cylinder, reducing to $\frac{3}{4}$ of its previous volume each stroke.

$Volumes: 1, \frac{3}{4}, \frac{9}{16}, \frac{27}{64}, \dots$

This is a geometric sequence, not an AP.
❌ Not an AP

(iii) Cost of digging a well increases by ₹50 for every meter.

$Costs: 150, 200, 250, 300, \dots$ $d = 200 - 150 = 50$

✅ AP with $a = 150, d = 50$ .

(iv) Amount under compound interest increases by 8% every year.

$A_{1} = P, A_{2} = P (1.08), A_{3} = P (1.08)^{2}, A_{4} = P (1.08)^{3}, \dots$

This is geometric, not arithmetic.
❌ Not an AP

Q2. Write the first four terms of each AP for the given $a$ and $d$ .

S.No	$a$	$d$	Terms
(i)	10	10	$10, 20, 30, 40$
(ii)	−2	0	$- 2, - 2, - 2, - 2$
(iii)	4	−3	$4, 1, - 2, - 5$
(iv)	−1	$\frac{1}{2}$	$- 1, - \frac{1}{2}, 0, \frac{1}{2}$
(v)	−1.25	−0.25	$- 1.25, - 1.50, - 1.75, - 2.00$

Q3. For each of the following APs, find the first term $a$ and common difference $d$ .

S.No	Sequence	$a$	$d$
(i)	$3, 1, - 1, - 3, \dots$	$3$	$- 2$
(ii)	$- 5, - 1, 3, 7, \dots$	$- 5$	$4$
(iii)	$\frac{1}{3}, \frac{5}{3}, 3, \frac{13}{3}, \dots$	$\frac{1}{3}$	$\frac{4}{3}$
(iv)	$0.6, 1.7, 2.8, 3.9, \dots$	$0.6$	$1.1$

Q4. Which of the following sequences are APs? If yes, find the common difference and three more terms.

(i) $2, 4, 8, 16, \dots$

$d_{1} = 4 - 2 = 2, d_{2} = 8 - 4 = 4$

Differences unequal → ❌ Not an AP

(ii) $\frac{5}{2}, \frac{7}{2}, \frac{9}{2}, \dots$

$d = \frac{7}{2} - \frac{5}{2} = 1$

✅ AP, next three terms: $\frac{11}{2}, \frac{13}{2}, \frac{15}{2}$

(iii) $- 1.2, - 3.2, - 5.2, - 7.2, \dots$

$d = - 3.2 - (- 1.2) = - 2.0$

✅ AP, next three: $- 9.2, - 11.2, - 13.2$

(iv) $- 10, - 6, - 2, 2, \dots$

$d = - 6 - (- 10) = 4$

✅ AP, next three: $6, 10, 14$

(v) Sequence unclear in print — possibly $3, 3 + 2, 3 + 4, 3 + 6, \dots$
If so:

$Differences 2, 2, 2 \Rightarrow d = 2$

✅ AP, next three: $3 + 8 = 11, 13, 15$
(Please confirm original question text for exact terms.)

(vi) $0.2, 0.22, 0.222, 0.2222, \dots$

$d_{1} = 0.02, d_{2} = 0.002, d_{3} = 0.0002$

Differences not equal → ❌ Not an AP

(vii) $0, - 4, - 8, - 12, \dots$

$d = - 4$

✅ AP, next three: $- 16, - 20, - 24$

(viii) $- \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, \dots$

$d = 0$

✅ AP, next three: $- \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}$

(ix) $1, 3, 9, 27, \dots$
Geometric, not arithmetic.
❌ Not an AP

(x) $a, 2 a, 3 a, 4 a, \dots$

$d = 2 a - a = a$

✅ AP, next three: $5 a, 6 a, 7 a$

(xi) $a, a^{2}, a^{3}, a^{4}, \dots$
Differences not equal (except for special $a$ values).
❌ Not an AP

(xii) $2, 8, 18, 32, \dots$

$d_{1} = 6, d_{2} = 10, d_{3} = 14$

Differences unequal → ❌ Not an AP

(xiii) $3, 6, 9, 12, \dots$

$d = 3$

✅ AP, next three: $15, 18, 21$

(xiv) $12, 32, 52, 72, \dots$

$d = 20$

✅ AP, next three: $92, 112, 132$

(xv) $12, 52, 72, 73, \dots$

$d_{1} = 40, d_{2} = 20, d_{3} = 1$

❌ Not an AP

✅ Summary Table of AP Sequences

Parts forming AP	Common difference (d)
(i) – ❌	—
(ii) – ✅	1
(iii) – ✅	−2
(iv) – ✅	4
(v) – ✅ (if as assumed)	2
(vi) – ❌	—
(vii) – ✅	−4
(viii) – ✅	0
(ix) – ❌	—
(x) – ✅	a
(xi) – ❌	—
(xii) – ❌	—
(xiii) – ✅	3
(xiv) – ✅	20
(xv) – ❌	—

Arithmetic Progressions (AP)

Q1. In which of the following situations does the list of numbers form an AP?

Q2. Write the first four terms of each AP for the given aa and dd.

Q3. For each of the following APs, find the first term aa and common difference dd.

Q4. Which of the following sequences are APs? If yes, find the common difference and three more terms.

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Q2. Write the first four terms of each AP for the given $a$ and $d$ .

Q3. For each of the following APs, find the first term $a$ and common difference $d$ .

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