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Sequences & Series: Distinct Odd Natural Numbers Divisible Prime Greater Less

JEE Maths question with a full step-by-step solution.

Question

a_1, a_2, \ldots, a_n

are distinct odd natural numbers not divisible by any prime greater than

5

, then

\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}

is less than

\frac{15}{8}

correct

\frac{16}{8}

\frac{8}{15}

\frac{15}{4}

Solution

Step 1: Each

a_i

is odd and uses only

3

and

5

, so

a_i=3^r 5^s

with

r,s\ge 0.

Step 2: The reciprocals are therefore bounded by the product of two geometric series:

\frac{1}{a_1}+\cdots+\frac{1}{a_n}<\left(1+\frac13+\frac{1}{3^2}+\cdots\right)\left(1+\frac15+\frac{1}{5^2}+\cdots\right)=\frac{1}{1-\frac13}\cdot\frac{1}{1-\frac15}=\frac32\cdot\frac54=\frac{15}{8}.

Step 3: Only finitely many terms are taken, so the sum is strictly less than

\dfrac{15}{8}.

Correct answer: (1)

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