Progression (Sequence and Series)Hard
Question
If a1, a2 ..... an are postive real number whose product is a fixed number c, them the minimum value of a1 + a2 + ...... + an-1 +2an is
Options
A.n(2c)1/n
B.(n + 1)c1/n
C.2nc1/n
D.(n + 1)(2c)1/n
Solution
Given, a1, a2 ..... an = c
⇒ a1a2a3 ...... (an - 1)(2an) = 2c .....(i)
∴
(a1.a2.a3 ...... 2an)1/n (using AM ≥ GM)
∴ a1 + a2 + a3 + ...... + 2an ≥ n(2c)1/n [from eq. (i)]
⇒ minimum value of
a1 + a2 + a3 + ...... + 2an = n(2c)1/n
⇒ a1a2a3 ...... (an - 1)(2an) = 2c .....(i)
∴
(a1.a2.a3 ...... 2an)1/n (using AM ≥ GM)∴ a1 + a2 + a3 + ...... + 2an ≥ n(2c)1/n [from eq. (i)]
⇒ minimum value of
a1 + a2 + a3 + ...... + 2an = n(2c)1/n
Create a free account to view solution
View Solution FreeMore Progression (Sequence and Series) Questions
If the sum of the first n natural numbers is 1/5 times the sum of the their squares, then the value of n is -...The geometric mean of the numbers 4, 8, 16 is-...The value of cot cot-1 is...If a,b,c are in A.P. and x = 1 + a + a2 + ...,y = 1 + b + b2 + ... and z = 1 + c + c2 + ...., (where a,b,c < 1)), the...Given two numbers a and b, let A denotes the single A.M. and S denote the sum of n A.M.′s between a and b, then S/...