Answer:
The total number of marbles in the set is 107 marbles
Step-by-step explanation:
The parameters given are;
Let the size of the marbles be 1 unit
The marbles are arranged to form the equilateral triangle as follows;
1st row = 1 marble
2nd row = 2 marbles and so on so that the total number of marbles for an arithmetic series as follows;
Total number of marbles, t = 1 + 2 + 3 + ... + n
Therefore;
Since there are 2 marbles left in forming the first equilateral triangle, we have the total number of marbles in the set, t₁, is given as follows;
When the same marble set are arranged into a triangle in which each side has one more marble than in the first arrangement, there where 13 marble shortage, hence, the total number of marbles is given as follows;
We therefore have;
Which gives;
Therefore;
n² + n + 4 = n² + 3·n - 24
2·n = 24 + 4 = 28
n = 14
From which we have;
Therefore, the total number of marbles in the set, t₁ = 107 marbles.