The minimum value of the sum of real numbers a–5, a–4, 3a

Engineering

Mathematics

Arithmetic Mean Geometric Mean and Harmonic Mean

Question

The minimum value of the sum of real numbers a^–5, a^–4, 3a^–3, 1, a⁸ and a¹⁰ with a > 0 is

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$A . M . \geq G . M .$

$\frac{a^{- 5} + a^{- 4} + a^{- 3} + a^{- 3} + a^{- 3} + 1 + a^{8} + a^{10}}{8} \geq {(a^{- 5} \cdot a^{- 4} \cdot a^{- 3} \cdot a^{- 3} \cdot a^{- 3} \cdot 1 \cdot a^{8} \cdot a^{10})}^{1 / 8}$

a^–5 + a^–4 + a^–3 + a^–3 + a^–3 + 1 + a⁸ + a¹⁰ $\geq$ 8

so minimum value is 8