JEE Main 2025 Apr 2 Shift 1, Mathematics Q24: Arithmetic, Geometric And Harmonic Progressions

JEE Main2025Apr 2, Shift 1Mathematics

Three distinct numbers are selected randomly from the set ${1, 2, 3, \dots, 40}$ . If the probability that the selected numbers are in an increasing G.P. with integer common ratio is $\frac{m}{n}$ , $g cd (m, n) = 1$ , then $m + n$ is equal to _____.

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Solution

Count the triples $(a, a r, a r^{2})$ that lie in ${1, 2, \dots, 40}$ , with $a \geq 1$ and $r > 1$ . Restricting to integer ratios $r$ :

$r = 2$ : $4 a \leq 40 \Rightarrow a \in {1, \dots, 10}$ , giving $10$ GPs.

$r = 3$ : $9 a \leq 40 \Rightarrow a \in {1, 2, 3, 4}$ , giving $4$ GPs.

$r = 4$ : $16 a \leq 40 \Rightarrow a \in {1, 2}$ , giving $2$ GPs.

$r = 5$ : $25 a \leq 40 \Rightarrow a = 1$ , giving $1$ GP.

$r = 6$ : $36 a \leq 40 \Rightarrow a = 1$ , giving $1$ GP.

$r \geq 7$ gives no valid GP.

Total favourable GPs $= 10 + 4 + 2 + 1 + 1 = 18$ .

Total ways to choose 3 distinct numbers (ordered as increasing GP corresponds to one unordered triple):

$(3 40) = 9880$

Required probability:

$P = \frac{18}{9880} = \frac{9}{4940}$

Since $g cd (9, 4940) = 1$ , $m = 9$ and $n = 4940$ .

$m + n = 9 + 4940 = 4949$

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JEE Main 2025 Apr 2 Shift 1, Mathematics Q24: Arithmetic, Geometric And Harmonic Progressions

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