A pilgrim starts walking for a journey of 115 km. On the first day he covers 7 km, the next day 9 km, and likewise keeps adding 2 km everyday till he reaches 15km per day which he maintains for the rest of the journey. How many days in all will he take to complete the journey?
Pilgrim's Journey Calculation
This problem involves calculating the total number of days a pilgrim takes to complete a 115 km journey, considering a specific pattern of daily distance covered.
Understanding the Daily Progress
The pilgrim's journey starts with a specific distance covered each day, and this distance increases over time. Let's break down the pattern:
- Day 1: Covers 7 km.
- Day 2: Covers 9 km (7 km + 2 km).
- Day 3: Covers 11 km (9 km + 2 km).
- Day 4: Covers 13 km (11 km + 2 km).
- Day 5: Covers 15 km (13 km + 2 km).
From Day 5 onwards, the pilgrim maintains a constant pace of 15 km per day until the journey is completed.
The distances covered during the first five days form an arithmetic progression: 7, 9, 11, 13, 15.
| Day | Distance (km) |
|---|---|
| 1 | 7 |
| 2 | 9 |
| 3 | 11 |
| 4 | 13 |
| 5 | 15 |
To find the total distance covered in these first 5 days, we can sum the terms of this arithmetic progression. The formula for the sum of an arithmetic progression ($S_n$) is given by:
$ S_n = \frac{n}{2} [2a + (n-1)d] $
Where:
- $n$ is the number of terms (days) = 5
- $a$ is the first term (distance on Day 1) = 7 km
- $d$ is the common difference (increase in distance each day) = 2 km
Calculating the sum for the first 5 days:
$ S_5 = \frac{5}{2} [2 \times 7 + (5-1) \times 2] $
$ S_5 = \frac{5}{2} [14 + (4) \times 2] $
$ S_5 = \frac{5}{2} [14 + 8] $
$ S_5 = \frac{5}{2} [22] $
$ S_5 = 5 \times 11 = 55 \text{ km} $
So, the pilgrim covers 55 km in the first 5 days.
Completing the Remaining Journey
The total journey is 115 km. After the first 5 days, the remaining distance is:
Remaining Distance = Total Journey - Distance Covered in First 5 Days
$ \text{Remaining Distance} = 115 \text{ km} - 55 \text{ km} = 60 \text{ km} $
From Day 6 onwards, the pilgrim covers 15 km each day. To find out how many more days are needed to cover the remaining 60 km, we divide the remaining distance by the daily distance covered:
Days for Remaining Journey = Remaining Distance / Daily Distance
$ \text{Days for Remaining Journey} = \frac{60 \text{ km}}{15 \text{ km/day}} = 4 \text{ days} $
Total Days Calculation
The total number of days taken to complete the journey is the sum of the days spent in the initial increasing phase and the days spent covering the remaining distance at a constant pace.
Total Days = Days in Initial Phase + Days for Remaining Journey
$ \text{Total Days} = 5 \text{ days} + 4 \text{ days} = 9 \text{ days} $
Therefore, the pilgrim will take a total of 9 days to complete the 115 km journey.
Important Questions from Progression (Notes)
- The sum of 16 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + .....$ is :
- A pilgrim starts walking for a journey of 115 km. On the first day he covers 7 km, the next day 9 km, and likewise keeps adding 2 km everyday till he reaches 15 km per day which he maintains for the rest of the journey. How many days in all will he take to complete the journey?
If a, b and c are in Geometric Progression and $a^\frac{1}{x} = b^\frac{1}{y} = c^\frac{1}{z}$ then, x, y, z are in ________.
Which of the following statement is true about the geometric series
$ 1 + r +r^2 + r^3 + ...............; (r > 0) $?
$6240$ रुपये की राशि $30$ किस्तों में इस प्रकार चुकाई जाती है कि प्रत्येक किस्त पिछली किस्त से $10$ रुपये अधिक है । पहली किस्त की मूल्य ____________है।
