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8. sequence and series — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS8. sequence and series1 Mark
MCQ
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}} + 2{a_n}$ is
  • ✓
    $n{(2c)^{1/n}}$
  • B
    $(n + 1)\,{c^{1/n}}$
  • C
    $2n{c^{1/n}}$
  • D
    $(n + 1){(2c)^{1/n}}$
✓

Answer

Correct option: A.
$n{(2c)^{1/n}}$
a
(a) $A.M.$ $ \ge $ $G.M. $

==> $\frac{{{a_1} + {a_2} + .... + {a_{n - 1}} + 2{a_n}}}{n} \ge {({a_1}.{a_{2,}}...{a_{n - 1}}2{a_n})^{\frac{1}{n}}} \ge {(2c)^{\frac{1}{n}}}$

Minimum value of ${a_1} + {a_2} + ...... + {a_{n - 1}} + 2{a_n} = n{(2c)^{1/n}}$.

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