A teacher is hosting an ever growing party of mucus people in his lungs.

On day one, there's 1 mucus person.
Day two, 4 more join the party.
Day three, 9 more come in, and on Day four, 16 more enter his lungs.

On the sixth day, the teacher decides to kick the party out using MucusB Gone, an over the counter medicine for ending mucus parties.

Assuming the medicine is completely effective, how many mucus people total will be eliminated?

Please, be sure to find the pattern rule, use sigma notation, and expand the notation to find your solution. Show ALL work and steps

To find the total number of mucus people eliminated by the medicine, let's first observe the pattern in the number of mucus people entering the lungs each day:

Day 1: 1 mucus person
Day 2: 4 mucus people
Day 3: 9 mucus people
Day 4: 16 mucus people

Notice that each day, the number of new mucus people is a perfect square:

Day 1: 1² = 1 mucus person
Day 2: 2² = 4 mucus people
Day 3: 3² = 9 mucus people
Day 4: 4² = 16 mucus people

Therefore, the pattern rule for the number of new mucus people entering the lungs (aₙ) on the nth day is:

[tex]a_n = n^2[/tex]

The sigma notation that represents the sum of the number of mucus people entering the lungs each day from day 1 to day 6 is:

[tex]\displaystyle \sum_{n=1}^{6} n^2[/tex]

Expanding this notation, we get:

[tex]\displaystyle \sum_{n=1}^{6} n^2=1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2[/tex]

Calculate the sum:

[tex]\displaystyle \sum_{n=1}^{6} n^2= 1 + 4 + 9 + 16 + 25 + 36 \\\\\\ \sum_{n=1}^{6} n^2 = 91[/tex]

So, the medicine eliminates a total of 91 mucus people from the teacher's lungs.