Exercise 5.1 Solutions

(i) The taxi fare after each km when the fare is ₹15 for the first km and ₹8 for each additional km.
Fare for 1st km = ₹15.
Fare for 2nd km = 15 + 8 = ₹23.
Fare for 3rd km = 23 + 8 = ₹31.
The list of fares is 15, 23, 31, …
The difference between consecutive terms is constant (8). Yes, this forms an AP.

(ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time.
Let the initial volume be V. After 1st removal: V – V/4 = (3/4)V.
After 2nd removal: (3/4)V – (1/4)(3/4)V = (9/16)V.
The list is V, (3/4)V, (9/16)V, …
The difference is not constant. No, this does not form an AP.

(iii) The cost of digging a well after every metre of digging, when it costs ₹150 for the first metre and rises by ₹50 for each subsequent metre.
Cost for 1st metre = ₹150.
Cost for 2nd metre = 150 + 50 = ₹200.
Cost for 3rd metre = 200 + 50 = ₹250.
The list of costs is 150, 200, 250, …
The difference between consecutive terms is constant (50). Yes, this forms an AP.

(iv) The amount of money in the account every year, when ₹10000 is deposited at compound interest at 8% per annum.
Amount after 1st year = 10000(1 + 8/100) = ₹10800.
Amount after 2nd year = 10800(1 + 8/100) = ₹11664.
The list is 10800, 11664, …
The difference is not constant (11664 – 10800 = 864, while the first increase was 800). No, this does not form an AP.

(i) a = 10, d = 10
Terms: 10, (10+10), (10+20), (10+30) => 10, 20, 30, 40.

(ii) a = -2, d = 0
Terms: -2, (-2+0), (-2+0), (-2+0) => -2, -2, -2, -2.

(iii) a = 4, d = -3
Terms: 4, (4-3), (4-6), (4-9) => 4, 1, -2, -5.

(iv) a = -1, d = 1/2
Terms: -1, (-1+1/2), (-1+1), (-1+3/2) => -1, -1/2, 0, 1/2.

(v) a = -1.25, d = -0.25
Terms: -1.25, (-1.25-0.25), (-1.25-0.50), (-1.25-0.75) => -1.25, -1.50, -1.75, -2.00.

(i) 3, 1, -1, -3, …
First term a = 3.
Common difference d = 1 - 3 = -2.

(ii) -5, -1, 3, 7, …
First term a = -5.
Common difference d = -1 - (-5) = 4.

(iii) 1/3, 5/3, 9/3, 13/3, …
First term a = 1/3.
Common difference d = 5/3 - 1/3 = 4/3.

(iv) 0.6, 1.7, 2.8, 3.9, …
First term a = 0.6.
Common difference d = 1.7 - 0.6 = 1.1.

(i) 2, 4, 8, 16, …
4 – 2 = 2; 8 – 4 = 4. Difference is not constant. Not an AP.

(ii) 2, 5/2, 3, 7/2, …
5/2 – 2 = 1/2; 3 – 5/2 = 1/2. Difference is constant. It is an AP.
d = 1/2. Three more terms: 4, 9/2, 5.

(iii) -1.2, -3.2, -5.2, -7.2, …
-3.2 – (-1.2) = -2. Difference is constant. It is an AP.
d = -2. Three more terms: -9.2, -11.2, -13.2.

(iv) -10, -6, -2, 2, …
-6 – (-10) = 4. Difference is constant. It is an AP.
d = 4. Three more terms: 6, 10, 14.

(v) 3, 3+√2, 3+2√2, 3+3√2, …
(3+√2) – 3 = √2. Difference is constant. It is an AP.
d = √2. Three more terms: 3+4√2, 3+5√2, 3+6√2.

(vi) 0.2, 0.22, 0.222, 0.2222, …
0.22 – 0.2 = 0.02; 0.222 – 0.22 = 0.002. Difference is not constant. Not an AP.

(vii) 0, -4, -8, -12, …
-4 – 0 = -4. Difference is constant. It is an AP.
d = -4. Three more terms: -16, -20, -24.

(viii) -1/2, -1/2, -1/2, -1/2, …
-1/2 – (-1/2) = 0. Difference is constant. It is an AP.
d = 0. Three more terms: -1/2, -1/2, -1/2.

(ix) 1, 3, 9, 27, …
3 – 1 = 2; 9 – 3 = 6. Difference is not constant. Not an AP.

(x) a, 2a, 3a, 4a, …
2a – a = a. Difference is constant. It is an AP.
d = a. Three more terms: 5a, 6a, 7a.

(xi) a, a², a³, a⁴, …
a² – a = a(a-1); a³ – a² = a²(a-1). Difference is not constant (unless a=1 or a=0). Not an AP.

(xii) √2, √8, √18, √32, …
Simplified: √2, 2√2, 3√2, 4√2, …
2√2 – √2 = √2. Difference is constant. It is an AP.
d = √2. Three more terms: 5√2 (√50), 6√2 (√72), 7√2 (√98).

(xiii) √3, √6, √9, √12, …
√6 – √3 ≠ √9 – √6. Difference is not constant. Not an AP.

(xiv) 1², 3², 5², 7², …
1, 9, 25, 49, …
9 – 1 = 8; 25 – 9 = 16. Difference is not constant. Not an AP.

(xv) 1², 5², 7², 73, …
1, 25, 49, 73, …
25 – 1 = 24; 49 – 25 = 24; 73 – 49 = 24. Difference is constant. It is an AP.
d = 24. Three more terms: 97, 121, 145.