Clara is 15 years old, and there are 120 coins on the box.
Each year her mom puts as many coins in her money box as how old she turns that day.
So on her first birthday, 1 coin is put in the box.
On her second birthday, 2 coins are put in the box
And so on.
So we have the simple summation:
[tex]\sum_{n = 1} n[/tex]
We know that the outcome of that summation is a number in the interval [110, 130]
If we know that the sum goes from n = 1 to n = N (N is the age of Clara) then we can rewrite the summation as:
[tex]N*(N + 1)/2[/tex]
Now we can solve:
[tex]N*(N + 1)/2 = 110\\\\N^2 + N = 220\\\\N^2 + N - 220 = 0[/tex]
The two solutions of the quadratic equation are:
[tex]N = \frac{-1 \pm \sqrt{1^2 - 4*1*(-220)} }{2} \\\\N = (-1 \pm 29.7) /2[/tex]
If we take only the positive solution:
N = (-1 + 29.7)/2 = 14.35
Now, notice that we got this number by taking the smallest possible number of coins (110) So we can round this to the next whole number, which is 15.
If N = 15, then the number of coins in the box is:
15*(15 + 1)/2 = 120
Which lies on the wanted interval.
If you want to learn more about summations:
https://brainly.com/question/542712
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