Arithmetic Progressions (AP)

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

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

$$\text{Fares: }15,23,31,39,\dots$$

Common difference:

$$d=23-15=8$$

✅ Yes, it is an AP with $a=15,d=8$.

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

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

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

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

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

✅ AP with $a=150,d=50$.

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(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

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

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

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

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

$$d_1=4-2=2,d_2=8-4=4$$

Differences unequal → ❌ Not an AP

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

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

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

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

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

✅ AP, next three: $-9.2,-11.2,-13.2$

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

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

✅ AP, next three: $6,10,14$

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

$$\text{Differences }2,2,2\Rightarrow d=2$$

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

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(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

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

$$d=-4$$

✅ AP, next three: $-16,-20,-24$

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

$$d=0$$

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

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

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

$$d=2a-a=a$$

✅ AP, next three: $5a,6a,7a$

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

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

$$d_1=6,d_2=10,d_3=14$$

Differences unequal → ❌ Not an AP

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

$$d=3$$

✅ AP, next three: $15,18,21$

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

$$d=20$$

✅ AP, next three: $92,112,132$

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

$$d_1=40,d_2=20,d_3=1$$

❌ Not an AP

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✅ Summary Table of AP Sequences