A boy is dividing M&Ms between himself and his sister. He gives 2 to his sister and takes 2 for himself. He gives 2 to his sister and takes 4 for himself. He gives 2 to his sister and takes 6 for himself, and so on.
(b) After n rounds, how many M&Ms does his sister have and how many does the boy have?
His sister has 2n
The boy has ? (2n+2n^2) confused on how to calculate his portion
sister has 2 n yes
boy has arithmetic sequence
a1 = 2
d = 2
an = a1 + (n-1) d = 2 + (n-1)2
or an = 2n
sum from n = 1 to n = n
S = (n/2)(a1+an) = (n/2 )(2 + 2n)
S = n + n^2
you got twice as many somehow
To calculate the number of M&M's the boy has after n rounds, we can set up a formula.
In the first round, the boy takes 2 M&M's for himself.
In the second round, he takes 4 M&M's.
In the third round, he takes 6 M&M's.
We can notice that the number of M&M's the boy takes in each round is increasing by 2. This is an arithmetic sequence with a common difference of 2.
To find the total number of M&M's the boy takes after n rounds, we can use the formula for the sum of an arithmetic sequence:
Sum = (n/2) * (first term + last term)
The first term is 2 (from the first round), and the last term is 2n (from the nth round). Plugging these values into the formula, we get:
Sum = (n/2) * (2 + 2n)
= (n/2) * (2n + 2)
= n(n + 1)
Therefore, after n rounds, the boy will have n(n + 1) M&Ms.
To calculate the boy's portion of M&Ms after n rounds, we can use the given information that he takes 2 more M&Ms for himself in each round.
Let's denote the number of M&Ms the boy has after n rounds as Bn.
In the first round, the boy takes 2 M&Ms for himself, so B1 = 2.
In the second round, he takes 4 more M&Ms for himself, so B2 = B1 + 4 = 2 + 4 = 6.
In the third round, he takes 6 more M&Ms for himself, so B3 = B2 + 6 = 6 + 6 = 12.
We can observe that the number of M&Ms the boy has after n rounds is increasing by 2 in each round and is given by the formula:
Bn = Bn-1 + 2n
Using this formula, we can calculate the boy's portion of M&Ms after each round:
B1 = 2
B2 = B1 + 2(2) = 6
B3 = B2 + 2(3) = 12
B4 = B3 + 2(4) = 20
And so on.
Therefore, the formula for the boy's portion of M&Ms after n rounds is:
Bn = 2n + 2n(n-1)
For example, if we want to find the boy's portion after 5 rounds, we plug in n=5 into the formula:
B5 = 2(5) + 2(5)(5-1) = 10 + 2(5)(4) = 10 + 2(20) = 10 + 40 = 50
So after 5 rounds, the boy will have 50 M&Ms.
Sister 6
Brother 12