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Proof by mathematical induction

If a1, a2,......

Question

If a1, a2,..............an are possitive real numbers whose product is a fixed number c, then the minimum value of a1+a2+.........a (n-1) + 2an

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Solution

Solution -: In this problem we aill apply AM -GM to the numbers a1, a2, a3,… ,an₋₁, 2an

(a1+a2+a3+… +an₋₁+2an) / n ≥ (a1a2a3… an₋₁*2an)^(1/n)

There is equality in AM-GM when all quantities are equal

→ (a1+a2+a3+… +an₋₁+2an) / n ≥ (2c)^(1/n) (equality if a1=a2…=an₋₁=2an)

→ a1+a2+a3+… +an₋₁+2an ≥ n(2c)^(1/n) so we will see that minimum value of

a1+a2+a3+… +an₋₁+2an is n(2c)^(1/n) so we got answer

n(2c)^(1/n) .

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