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CAT 2022 Slot 2 — QA Question 20

Miscellaneous ProgressionsEasy

On day one, there are 100 particles in a laboratory experiment. On day n, where n greater than or 2, one out of every n particles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals.

Answer & solution

  • A

    16

  • B

    17

  • 19

  • D

    18

Solution

Easy

On day nn, one in every nn particles doubles, so the count multiplies by n+1n\frac{n+1}{n}. The product telescopes into a clean linear formula in mm.

1

Daily multiplier. On day nn, nn particles become n+1n+1, so:

Pn=Pn−1×n+1nP_n=P_{n-1}\times\dfrac{n+1}{n}
2

Telescope from day 2 up to day mm (start P=100P=100):

Pm=100⋅32⋅43⋅54⋯m+1m=100⋅m+12P_m=100\cdot\dfrac{3}{2}\cdot\dfrac{4}{3}\cdot\dfrac{5}{4}\cdots\dfrac{m+1}{m}=100\cdot\dfrac{m+1}{2}
3

Set equal to 1000 and solve:

100⋅m+12=1000 ⇒ m+12=10 ⇒ m+1=20 ⇒ m=19100\cdot\dfrac{m+1}{2}=1000\ \Rightarrow\ \dfrac{m+1}{2}=10\ \Rightarrow\ m+1=20\ \Rightarrow\ m=19
m=19m=\mathbf{19}

The telescoping leaves only the first denominator 22 and the last numerator m+1m+1, so the count is just 100⋅m+12=50(m+1)100\cdot\frac{m+1}{2}=50(m+1). Need 50(m+1)=1000⇒m=1950(m+1)=1000\Rightarrow m=19.

CAT 2022 Slot 2 QA Q20: On day one, there are 100 particles in a laboratory experiment. On day n, where n greater than or 2, one out o — Solution | TheCATExam