The taxi fare after each km, when the fare is Rs 15 for the first km and Rs 8 for each additional km, does not form an AP as the total fare (in Rs) after each km is \(15, 8, 8, 8, \ldots\). Is this statement true? Give reasons.
False. The total fare after each km is \(15, 23, 31, 39, \ldots\), which is an AP with \(a=15\), \(d=8\).
Let us carefully calculate the fare step by step.
Step 1: For the first kilometre, the fare is Rs 15. So, the total after 1 km = 15.
Step 2: For the second kilometre, we add Rs 8 more. So, total after 2 km = 15 + 8 = 23.
Step 3: For the third kilometre, again add Rs 8. So, total after 3 km = 23 + 8 = 31.
Step 4: For the fourth kilometre, again add Rs 8. So, total after 4 km = 31 + 8 = 39.
So the total fares after each km are: \(15, 23, 31, 39, \ldots\).
Step 5: Now check if this is an Arithmetic Progression (AP). In an AP, the difference between two consecutive terms must be the same.
Here: \(23 - 15 = 8, \; 31 - 23 = 8, \; 39 - 31 = 8\). The difference is always 8.
Conclusion: The total fares form an AP with first term \(a = 15\) and common difference \(d = 8\). So the given statement is False.