Question
If a1, a2, a3, ........., an are positive real numbers whose product is a fixed number c, then the minimum value of a1 + a2 + a3 + .... + an – 1 + 2an is
- n(2c)1/n
- (n + 1) c1/n
- 2nc1/n
- (n + 1)(2c)1/n
Hint:
Use the A.M- G.M. relationship of sequences
The correct answer is: n(2c)1/n
n(2c)1/n
Given that a1.a2.a3.a4…..an = c
From the AM GM relationship, we know that AM>=GM
Or
na1+a2+...+an−1+2an ≥ (a1a2...an−1(2an))n1
substituting the value of c , we get
a1 + a2 + ….2an >= n(2c)^(1/n)
therefore, the minimum value of a1+a2+a3+….2an = n(2c)^(1/n)
the Am- Gm relation is a handy tool for solving sequence related problems. it is applicable to any sequence of numbers.
Related Questions to study
Let 
and 
are two arithmetic sequences such that 
and 
then the value of 
is
an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
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an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
If a, b and c are three consecutive positive terms of a G.P. then the graph of y = ax2 + bx + c is
Nature of graph of a quadratic equation is given by its discriminant.
Here, a = a, b= ar, c = ar2 , where r is the common ratio of the GP
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If a, b and c are three consecutive positive terms of a G.P. then the graph of y = ax2 + bx + c is
Nature of graph of a quadratic equation is given by its discriminant.
Here, a = a, b= ar, c = ar2 , where r is the common ratio of the GP
Given, a,b,c>0
If ln (a + c) , ln (c – a), ln (a – 2b + c) are in A.P., then :
an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
An = A1+(n-1)x d
A2= A1 + d
A3 = A1 + 2d
A3 = A2+d
A3-A2=A2-A1
If ln (a + c) , ln (c – a), ln (a – 2b + c) are in A.P., then :
an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
An = A1+(n-1)x d
A2= A1 + d
A3 = A1 + 2d
A3 = A2+d
A3-A2=A2-A1
