JEE Main 2019 — Sequences And Series Question with Solution

As triangle is equilateral so each side of triangle has n-balls.

At the top of the triangle there is 1 ball then next line has 2 balls. Similarly last line has n balls.

$$\therefore$$ Total no of balls used to create this triangle

= 1 + 2 + 3 + ..............+ n

= $$\frac{n\left(n+1\right)}{2}$$
Here Number of balls in each side of square is = (n – 2)

$$\therefore$$ Total no of balls used to create this square

= (n – 2)²

According to the question,

$$\frac{n\left(n+1\right)}{2}$$ + 99 = (n – 2)²

$$\Rightarrow$$ n² - 9n - 190 = 0

$$\Rightarrow$$ (n - 19) (n + 10) = 0

$$\therefore$$ n = 19

So total balls used to form triangle

= $$\frac{n\left(n+1\right)}{2}$$ = $$\frac{19\left(20\right)}{2}$$ = 190