Question

Divakar and Subhank set out to meet each other from Deli and Saharanpur, \(330 \mathrm{~km}\) apart. Divakar travels \(30 \mathrm{~km}\) on the first day \(28 \mathrm{~km}\) on the second day, \(26 \mathrm{~km}\) on the third day and so on. Subhank travels \(20 \mathrm{~km}\) on the first day, \(24 \mathrm{~km}\) on the second day, \(28 \mathrm{~km}\) on the third day and so on. In how many days they meet, if they started moving towards each other at the same time ? (a) 4 (b) 5 (c) 6 (d) none of these

Short answer

a) 4 b) 5 c) 6 d) 7 Answer: (b) 5

Step-by-step solution

Model the distances traveled by Divakar and Subhank as arithmetic sequences

Divakar's daily distance sequence: D = 30 - 2(n - 1) for n ≥ 1 \\ Subhank's daily distance sequence: S = 20 + 4(n - 1) for n ≥ 1

Calculate the sum of their daily distances

Sum of the first n terms of Divakar's sequence: SD = n(2 * 30 - 2(n - 1))/2 = n(60 - 2n + 2) \\ Sum of the first n terms of Subhank's sequence: SS = n(2 * 20 + 4(n - 1))/2 = n(40 + 4n - 4)

Set up an equation for the sum of their daily distances equal to 330 km

The equation will be: SD + SS = 330 \\ n(60 - 2n + 2) + n(40 + 4n - 4) = 330

Simplify the equation

Expanding and simplifying the equation: \\ n(60 - 2n + 2 + 40 + 4n - 4) = 330 \\ n(98 + 2n) = 330

Solve the equation for n

Divide both sides by 2: \\ n(49 + n) = 165 \\ 49n + n^2 = 165 \\ n^2 + 49n - 165 = 0

Find the integer solutions for n

The equation we derived is a quadratic equation, but since we are looking for the number of days, we want integer solutions. By factoring the equation or using the quadratic formula, we can find the possible integer solutions for n: \\ n = 5, -33 \\ Since the number of days cannot be negative, n = 5. Answer: (b) 5

Key concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers where each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This simple definition is crucial when modeling problems where quantities increase or decrease at a regular rate.

In the exercise, Divakar and Subhank's travel distances are examples of arithmetic sequences with common differences of -2 km/day and +4 km/day, respectively. Such sequences allow us to predict and calculate future values effectively. An AP is defined by its initial term, and the common difference, allowing us to express any term in the sequence as follows: \( a_n = a_1 + (n - 1) \cdot d \) where \( a_n \) is the n-th term, \( a_1 \) is the first term, and \( d \) is the common difference. The formula can be thought of as jumping steps, where you start at the first number and take leaps, defined by the common difference, to reach subsequent numbers.

Distance Calculation

Distance problems often involve calculating how far someone or something travels over time. We use arithmetic progressions to model situations where travel occurs at varying rates that change at a constant amount.

For Divakar and Subhank, we see their distances forming two separate sequences. To solve for the time it takes them to meet, we have to find the point where the sum of their traveled distances equals the distance between them. This application shows how arithmetic sequences can be powerful tools in calculating distances over time under conditions of constant change. When solving such problems, it's essential to accurately model the scenario with the right sequences, interpreting the common difference as the rate of change for each subject or object in motion.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x with a standard form of \( ax^2 + bx + c = 0 \) where \( a \) is not equal to 0. To solve quadratic equations, we can factorize, complete the square, or use the quadratic formula, which provides the solutions as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) .

In our particular problem, the sum of the distances traveled by Divakar and Subhank leads us to a quadratic equation. While the general solution to these equations may yield both positive and negative values, in the context of this exercise, only the positive value makes sense as it would not be logical to have negative days. Therefore, we discard the negative solution and accept the positive one as the answer to our problem.

Sequence Sum

The sum of a sequence, particularly an arithmetic sequence, is the total value when all terms of the sequence are added together. For arithmetic progressions, the sum of the first n terms can be calculated using the formula: \( S_n = \frac{n}{2}(a_1 + a_n) \) or alternatively \( S_n = \frac{n}{2}(2a_1 + (n - 1) \cdot d) \) where \( S_n \) is the sequence sum, \( n \) is the number of terms, \( a_1 \) is the first term, \( a_n \) is the n-th term, and \( d \) is the common difference.

This formula is derived from the idea that the sum of the sequence is equivalent to the sum of pairs of terms equidistant from the start and the end of the sequence. In the context of our exercise, we calculated the sequence sums of both Divakar’s and Subhank’s distances to find the point at which they converge; this is a practical application of the concept in solving real-world problems.