Question

Two brothers are dividing M\&Ms between themselves. The older one gives one to his younger brother and takes one for himself. He gives another to his younger brother and takes two for himself. He gives a third one to younger brother and takes three for himself, and so on. (a) On the \(n^{\text {th }}\) round, how many M\&Ms does the older boy give his younger brother? How many does he take himself? (b) After \(n\) rounds, how many M\&Ms does his brother have? How many does the older boy have?

Short answer

Calculate the number of M&M's each brother has after n rounds. Answer: In the nth round, the older brother gives 1 M&M to the younger brother and takes n M&M's for himself. After n rounds, the younger brother has a total of n M&M's, and the older brother has \(\frac{n(n+1)}{2}\) M&M's.

Step-by-step solution

(Part a: M&M's in the nth round)

Firstly, notice that the older brother gives 1 M&M to his younger brother in each round, so on the nth round, he gives the younger brother 1 M&M as well. For himself, he takes the same number of M&m's as the number of the round they are in, so he takes n M\&Ms in the nth round.

(Part b: M&M's after n rounds for the younger brother)

To calculate the total number of M&M's the younger brother has after n rounds, we need to add the M&M's he received in each round. Since he gets 1 M&M in each round, after n rounds, he will have a total of n M&M's.

(Part b: M&M's after n rounds for the older brother)

To find the total number of M&M's the older brother has after n rounds, we need to add the M&M's he took for himself in each round. In the first round, he took 1 M&M, in the second round he took 2 M&M's, and so on. In the nth round, he took n M&M's. We can represent the total number of M&M's he has after n rounds as the sum of the arithmetic series: \(S_n = 1+2+3+...+n\). This can be calculated using the formula for the sum of an arithmetic series: \(S_n = \frac{n(n+1)}{2}\). So, the older brother has \(\frac{n(n+1)}{2}\) M&M's after n rounds.

Key concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics, enabling us to solve problems by representing numbers and operations in a concise manner. In this exercise, the older brother decides to distribute M&Ms to his younger sibling following a set pattern. The algebraic expression helps us represent this pattern succinctly.

For instance, in each round, we observe the older sibling always gives one M&M to the younger brother. Therefore, the number of M&Ms the younger brother receives after each round can be expressed straightforwardly as the expression: \(1 \times n\). Here, \(n\) signifies the number of rounds. Such an expression allows for easy computation without recalculating at each step.

Similarly, the older brother takes M&Ms for himself based on the round number, leading us to the expression \(n\) for the \(n\)th round. This systematic approach makes it simple to replace \(n\) with any round number to calculate the M&Ms distributed in each round.

Problem Solving

Problem solving is an analytical skill that enables us to break down a complex scenario and find an efficient solution. This exercise highlights several elements of problem solving through patterns and sequences. By analyzing the problem, we can discern the consistent action the older brother undertakes: giving one M&M and taking a number corresponding to the round.

To solve part (a) of the problem, we recognize that in each round, the older brother gives exactly 1 M&M, while for himself, he takes M&Ms equivalent to the current round number. Recognizing this pattern simplifies our computational task immensely.

As we progress to part (b), the problem-solving technique involves compiling these patterns over several rounds. Specifically, by summing the consistent values over \(n\) rounds for both brothers, we effectively solve the question—illustrating that breaking down a problem into repeatable units helps in managing larger calculations.

Arithmetic Sequences

Arithmetic sequences form the basis of patterns where the difference between consecutive terms is constant. In this scenario, the sequence of M&Ms taken by the older brother forms an arithmetic sequence. This type of sequence helps us quickly compute sums over a series of rounds.

The older brother's pattern of taking M&Ms follows the arithmetic sequence \(1, 2, 3, \ldots, n\). This sequence simplifies our task to find the total M&Ms he possesses after \(n\) rounds by employing the formula for the sum of an arithmetic sequence: \(S_n = \frac{n(n+1)}{2}\).

Understanding this formula is crucial as it takes into account every incremental addition without manually adding each value. Arithmetic sequences play a pivotal role when dealing with incrementally increasing distributions—as seen with how the older brother takes M&Ms. This reduces complex addition to a simple formula application.