A person books a taxi to travel in the city. The taxi company charges a fixed booking fee of ₹200 plus ₹40 per kilometre travelled. Let us write the sequence representing the total fare after travelling 1 km, 2 km, 3 km, and so on. If the person travels 10 km, what will be the total fare?
Step-by-step solution
To find: The sequence of fares for 1, 2, 3, … km, and the fare for 10 km
Idea: Each extra kilometre adds the same ₹40, so the fares form an AP. Use a + (n − 1)d for the fare after n km.
- After 1 km: 200 + 40 = ₹240; after 2 km: 200 + 80 = ₹280; after 3 km: 200 + 120 = ₹320.½ mark
- 240, 280, 320, … is an AP with a = 240 and d = 40.½ mark
- tn = 240 + (n − 1) × 40 = 240 + 40n − 40 = 200 + 40n, where n is the distance in km.1 mark
- For 10 km: t10 = 200 + 40 × 10 = 200 + 400 = ₹600.1 mark
Check: Directly: booking fee 200 + 10 km × 40 = 200 + 400 = ₹600 ✓.
Answer to write in the exam
After 1 km: 200 + 40 = 240; 2 km: 280; 3 km: 320
Sequence 240, 280, 320, …: AP, a = 240, d = 40
tn = 240 + (n − 1) × 40 = 200 + 40n
t10 = 200 + 40 × 10 = 600
∴ Total fare for 10 km = ₹600
Common mistakes that cost marks
- Starting the AP at 200. ₹200 is the fare for 0 km; the fare after 1 km is ₹240.
- Using 240 + 40 × 10 = ₹640 (forgetting the −1 in n − 1).
- Thinking the fare doubles with distance. Only the ₹40 part grows; the ₹200 is paid once.
How this can come in the exam
An auto-rickshaw charges ₹30 for the first kilometre and ₹15 for every further kilometre.
(i) Write the fares for 1, 2, 3, 4 km. (ii) Show that they form an AP and write the fare for n km. (iii) Find the fare for 12 km. (iv) How far can you travel for ₹225?
Show answer
(i) ₹30, ₹45, ₹60, ₹75 (1 mark). (ii) Difference 15 each time, so AP with a = 30, d = 15; fare = 30 + 15(n − 1) = 15n + 15 (1 mark). (iii) 15 × 12 + 15 = ₹195 (1 mark). (iv) 15n + 15 = 225 ⇒ n = 14 km (1 mark).Try one yourself
A gym charges a joining fee of ₹500 and ₹800 per month. Write the total paid after 1, 2, 3 months and find the total after 12 months.
Show answer
₹1300, ₹2100, ₹2900, … (AP, d = 800). After n months: 500 + 800n; after 12 months: ₹10,100.
More questions like this
- Can you find the sum of the first ten natural numbers without actually adding all of them?
- Can the same approach be used to find the sum of 1 + 2 + 3 + … + 100?
- If we use the notation Sn to represent the sum of the first n natural numbers, then Sn = n(n + 1)2. Can you use this formula to find S20, S50 or S1000?
- Let us revisit the sequence tn of triangular numbers 1, 3, 6, 10, 15, … shown in the figure. Note that the nth term of this sequence is the sum of the first n natural numbers. Thus tn = n(n + 1)2. Can you use this to find the 10th, 17th and 80th triangular numbers?
- Find the 10th and 26th terms of the AP: 3, 8, 13, 18, ….
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