Mathematic question

Question

A group of students collected 200 seashells at a beach. What would be the maximum possible

number of students in the group, if every student collected at least one seashell, and all

students collected different numbers of seashells?

Answer is 19, need the detailed solution.

Answer 1

Let us keep adding the shells as follows:

1st student collected 1 shell

2nd student collected 2 shells

2 students together collected 1+ 2 = 3 shells

3rd student collected 3 shells

3 students together collected 3 + 3 = 6 shells

4th student collected 4 shells

4 students together collected 6 + 4 = 10 shells

5th student collected 5 shells

5 students together collected 10 + 5 = 15 shells

6th student collected 6 shells

6 students together collected 15 + 6 = 21 shells

…

Let us see whether we can figure out the pattern.

1 = (1² + 1)/2

3 = (2² + 2)/2

6 = (3² + 3)/2

…

21 = (6² + 6)/2

…

so to find the number of shells you need to square the number of students add the number of students and divide the sum by 2 (this can be proven by induction, please, comment if you need the proof, and I will add it)

now we need to solve the inequality

(n² + n)/2 ≤ 200, where n is a positive integer

n² + n ≤ 400

n ≤ 400 - n²

n ≤ (20 - n)(20 + n), clearly n cannot be 20 or higher because the expression on the right is zero for n = 20 and negative for n > 20

However n can be any integer number less than 20 (namely, 19 or below), because if 1 ≤ n ≤ 19, then (20-n) ≥ 1 and (20+n) ≥ 21, therefore

(20 - n)(20 + n) ≥ 21 > 19 ≥ n

So if there are 19 students, one possible way (there are others too) for them to collect 200 shells would be for the first 18 students to collect (18² + 18)/2 = 171 shells and the 19th student to collect 29 shells (200 - 171 = 29)

Answer 2

You can "crunch" the numbers with addition

1 + 2 = 3

3 + 4 = 7

7 + 5 = 12

12 + 6 = 18

18 + 7 = 25

15 + 8 = 33

How "far" must you ge to get to 200?

Answer 3

200 shells

each student collected at least one

each student collected a different number from other students

what is the maximum number of students? 19

1,2,3,4,5,... is an arithmetic sequence with common difference, d, = 1, a1, first term = 1

sum of a partial arithmetic sequence =

Sn = (n/2)(a1+an) = 200

an = n, a1=1

solve for n

400 = (n)1 +n^2

n^2 + n - 400 = 0

use the quadratic formula

n =-1/2 + or - (1/2)sqr(1+1600))

n =about -1/2 +20

n= about 19 1/2

has to be an integer

19 is the maximum number of students