Progressions:The Non-routine Way
Algebra
Level
3
Five distinct two-digit natural numbers are in geometric progression. What is the middle term?
The answer is 36.
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1 solution
As there are 5 terms of a G.P. and all of them are two digit numbers, the common ratio cannot be a whole number. Hence the common ratio is a fraction. Now the first term should be the fourth power of the denominator of the common ratio. The only two-digit numbers which are fourth powers are 16 and 81. If the common ratio is of the form 2 p then the numerator should be an odd number. If we take numerator greater than 3, then the condition of 2-digit numbers is not satisfied.Hence numerator is 3. So the sequence becomes 16,24,36,54,81. If the denominator is taken 3, then by the same reasoning, we get the sequence 81,54,36,24,16. In both the cases, the middle term is 36, hence the answer is 36.