If a_1,a_2, ... ... .,a_n are positive real numbers whose product is a fixed number c then the minimum value of a_1 + a_2 + ... ... + a_n - 1 + 2a_n is

(a) : Given a_1 * a_2 * a_3 ... ... . a_n=c \ => a_1 * a_2 * a_3 ... ... . . a_n-1(2 a_n)=2 c\ We know that A.M. \ >= G.M. A.M. of a_1, a_2, ... ... , 2 a_n is (a_1+a_2+a_3+2 a_n / n) G.M. of a_1, a_2, a_3, ... ... , 2 a_n is (a_1 a_2 a_3 ... ... ... .2 a_n)^1 / n ⇒ G.M. =(2 c)^1 / n Now, A.M. >= G.M. => a_1+a_2+a_3+ * s ... +2 a_n >= n(2 c)^1 / n ⇒ minimum value of (a_1+a_2+a_3+ ... . .+2 a_n)=n(2 c)^1 / n

If a_{1},a_{2},.,a_{n} are positive real numbers whose ... | Sequence & Series (Mathematics)

(a) : Given $a_1 \cdot a_2 \cdot a_3 \ldots \ldots . a_n=c$ $\\ \Rightarrow a_1 \cdot a_2 \cdot a_3 \ldots \ldots . . a_{n-1}\left(2 a_n\right)=2 c\\$ We know that A.M. $\\ \geq$ G.M. A.M. of $a_1, a_2, \ldots \ldots, 2 a_n$ is $\frac{a_1+a_2+a_3+2 a_n}{n}\\$ G.M. of $a_1, a_2, a_3, \ldots \ldots, 2 a_n$ is $\left(a_1 a_2 a_3 \ldots \ldots \ldots .2 a_n\right)^{1 / n}\\$ $\text { ⇒ G.M. }=(2 c)^{1 / n}\\$ Now, A.M. $\geq$ G.M. $\Rightarrow a_1+a_2+a_3+\cdots \ldots+2 a_n \geq n(2 c)^{1 / n} \\$ ⇒ minimum value of $\left(a_1+a_2+a_3+\ldots . .+2 a_n\right)=n(2 c)^{1 / n}$

Explanation

Let a_{1},a_{2},,a_{n} be a given A.P. whose common difference is an integer and S_{n} = a_{1} + a_{2} + + a_{n}. If a_{1} = 1,a_{n} = 300 and 15 \leq n \leq 50, then the ordered pair ( S_{n - 4},a_{n - 4} ) is equal to

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