Five distinct 2 digit number are in geometric progression. Find the sum of second and fourth digit
Answers
Given : Five distinct 2digit no are in a geometric progression.
To find : sum of second and fourth digit
Solution:
Let say
Five Digits are
a , ar , ar² , ar³ , ar⁴
ar⁴ < 100
a ≥ 10
=> r⁴ < 10
=> r < 1.78
r ≠ 1
So r will be a rational number Let say (p/q) where p & q are co prime & integers
now a (p/q)⁴ is an integer
=> a must be = kq⁴ k is an integer ( as p & q are co prime)so q⁴ in denominator must be cancelled by a
Hence a can be k(2)⁴ or k (3)⁴ because 4⁴ > 99 so q can be 2 or 3 only
Hence a can be 16k or 81k
Case 1 :
q = 2
=> a = 16k then a (p/q)⁴ = kp⁴ then p can be 3 only & k has to be 1
k > 1 will make ar⁴ > 99
so r = p/q = 3/2
a = 16 as k = 1
16 , 24 , 36 , 54 , 81 are the number
if q = 3
then a = 81k hence k must be 1 => a = 81
a (p/q)⁴ = p⁴ Then p must be 2 as q = 3
so r = 2/3
Numbers
are 81 , 54 , 36 , 24 , 16
Sum of 2nd & fourth Digit in both cases = 24 + 54 = 78
78 is the sum of second and fourth digit
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