Question

a, b, c, d and e are integers such that $1 \le a < b < c < d < e$. If a, b, c, d and e are in geometric progression and $	ext{lcm}(m, n)$ is the least common multiple of $m$ and $n$, then the maximum value of $$\frac{1}{	ext{lcm}(a,b)} + \frac{1}{	ext{lcm}(b,c)} + \frac{1}{	ext{lcm}(c,d)} + \frac{1}{	ext{lcm}(d,e)}$$ is

Accepted Answer

B. Let the GP be $a, ar, ar^2, ar^3, ar^4$ with common ratio $r > 1$ integer. $	ext{lcm}(ar^k, ar^{k+1}) = ar^{k+1}$. The sum becomes $\frac{1}{a} \left( \frac{1}{r} + \frac{1}{r^2} + \frac{1}{r^3} + \frac{1}{r^4} ight)$. To maximize this expression, we take the minimum possible integer values for $a$ and $r$: $a = 1$ and $r = 2$. Max value = $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = \frac{15}{16}$.

GP LCM Maximum Sum

Explanation

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