JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Sequences and Series

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Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

(A)190✓ Correct
(B)262
(C)225
(D)157

Answer: (A)

Solution

Let the triangle have

n

rows, so it uses

1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}

balls. Adding

99

balls lets them fill a square whose side holds two balls fewer than the triangle's side, that is

(n - 2)

per side, so

\frac{n ( n + 1 )}{2} + 99 = (n - 2)^{2}

. Clearing the fraction,

n^{2} + n + 198 = 2 n^{2} - 8 n + 8

, which tidies to

n^{2} - 9 n - 190 = 0

, or

(n - 19) (n + 10) = 0

, giving

n = 19

. The triangle therefore holds

\frac{19 \times 20}{2} = 190

balls.

JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Single correct option

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JEE Main · 2019 · 09 Apr 2019 · Shift 2 · Sequences and Series

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