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If \[abcd{\text{ }} = {\text{ }}256\] where a, b, c and d are positive numbers, then the minimum value is
\[\left( {a + b + c + d} \right)\]
Option:
A). \[4\]
B). \[8\]
C). \[16\]
D). \[64\]

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Last updated date: 23rd Jul 2024
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Answer
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Hint: The question is taken from the series and sequence and sub part of the relation between the arithmetic progression (A.P.) and geometrical progression (G.P.). (A.P.) is a sequence of numbers in order in which the difference of two consecutive numbers is always the same and (G.P.) is a sequence of numbers in order in which the succeeding term is given by the multiplication of each preceding term by the constant number. Consider a, b, c, d are in A.P and G.P. then we can use the relation below to solve the question so
\[a + b + c + d \geqslant {\left( {abcd} \right)^{\dfrac{1}{n}}}\]
Where n is number of the terms

Complete step-by-step solution:
Step 1:
First, we write the given data
\[abcd{\text{ }} = {\text{ }}256\]
Now put the values into the above formula which is mentioned in the hint here we consider the a, b, c, d are in the A.P. and the G.P. so the relation is governed between two is given by below
\[a + b + c + d \geqslant {\left( {abcd} \right)^{\dfrac{1}{n}}}\]
Now put the given value in the formula there is four term so \[n = 4\] and \[abcd{\text{ }} = {\text{ }}256\] then
\[a + b + c + d \geqslant {\left( {abcd} \right)^{\dfrac{1}{n}}}\]
\[\Rightarrow a + b + c + d \geqslant {\left( {256} \right)^{\dfrac{1}{4}}}\]
\[\Rightarrow a + b + c + d \geqslant {\left( {{2^8}} \right)^{\dfrac{1}{4}}}\]
\[\Rightarrow a + b + c + d \geqslant \left( {{2^{8 \times \dfrac{1}{4}}}} \right)\]
\[\Rightarrow a + b + c + d \geqslant \left( {{2^2}} \right)\]
\[\Rightarrow a + b + c + d \geqslant \left( 4 \right)\]
\[\Rightarrow \left( {a + b + c + d} \right) \geqslant 4\]
Now the minimum value of the \[\left( {a + b + c + d} \right)\] is given by the follow
\[{\left( {a + b + c + d} \right)_{\text{minimum}}} = 4\]
Hence we got the minimum value of the given term which comes to the 4 and further extend hence the correct option from the given option is (1).

Note: The question is based on the Sequence and series and the subpart of the arithmetic progression and geometric progression G.P. relation and we can find any minimum value of the given sequence by using the above formula when the sum of the term is given and as well we can find the maximum value of the product of the term using the formula.