a,b,c,d and e are integers such that 1
≤ a < b < c < d < e. If a,b,c,d and e are in geometric progression
and lcm(m,n) is the least common multiple of m and n, then the maximum value of
1/lcm(a,b) + 1/lcm(b,c) + 1/lcm(c,d) + 1/lcm(d,e) is:

(A)1 (B) 15/16 (C) 79/81 (D) 7/8 (E) None of these

(A)1 (B) 15/16 (C) 79/81 (D) 7/8 (E) None of these

Answer follows here:

Given that the numbers are in G.P.

Let the common ratio be ‘r’, hence
the series a,b,c,d,e can also be expressed as:

a,ar,ar

^{2},ar^{3},ar^{4}
lcm(a,b) = lcm(a,ar) = ar

lcm(b,c) = lcm(ar,ar

^{2}) = ar^{2}
lcm(c,d) = lcm(ar

^{2},ar^{3}) = ar^{3}
lcm(d,e) = lcm(ar

^{3},ar^{4}) = ar^{4}
1/lcm(a,b) + 1/lcm(b,c) + 1/lcm(c,d)
+ 1/lcm(d,e)

= 1/ar + 1/ar

^{2}+ 1/ar^{3}+ 1/ar^{4}
= 1/a (1/r + 1/r

^{2}+ 1/r^{3}+ 1/r^{4}) --------(1)
To get max value of this, ‘a’ and ‘r’
should be minimum.

It is given that 1 ≤ a => Minimum value
of ‘a’ = 1

For the values in the series to be
integers, the minimum common ratio,

r = 2

r = 2

(r ≤ 1 won’t work here as it is an increasing GP)

Substituting ‘a’ and ‘r’ values in
the expression (1),

Maximum value = 1/2+ 1/4 + 1/8 + 1/16
= (8+4+2+1)/16 = 15/16

**Answer (B)**

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