If a1, a2, a3 ....... an are positive real numbers whose product is a

1.	If a1, a2, a3 ....... an are positive real numbers whose product is a fixed number ‘c’, then the minimum value of a1 + a2 ..... + an–1 + 2an is(a) n (2c)1/n (b) (n + 1)c1/n (c) 2nc1/n (d) (n + 1) (2c)1/n
Answer» (c) n(2c)^\(\frac1n\) For positive real numbers, AM > GM ⇒ \(\frac{(a_1+a_2+a_3+.......+2a_n)}{n}\) > (a₁.a₂.a₃.......2a_n)^\(\frac1n\) ⇒ (a₁ + a₂+ a₃+......+ 2a_n) > n(c.2)^\(\frac1n\) = n(2c)^{^\(\frac1n\)} Hence the least value is n(2c)^\(\frac1n\)

If a1, a2, a3 ....... an are positive real numbers whose product is a fixed number ‘c’, then the minimum value of a1 + a2 ..... + an–1 + 2an is(a) n (2c)1/n (b) (n + 1)c1/n (c) 2nc1/n (d) (n + 1) (2c)1/n

Answer»

(c) n(2c)^\(\frac1n\)

For positive real numbers, AM > GM

⇒ \(\frac{(a_1+a_2+a_3+.......+2a_n)}{n}\) > (a₁.a₂.a₃.......2a_n)^\(\frac1n\)

⇒ (a₁ + a₂+ a₃+......+ 2a_n) > n(c.2)^\(\frac1n\) = n(2c)^{^\(\frac1n\)}

Hence the least value is n(2c)^\(\frac1n\)