induction with k parameters (1 Viewer)

bobohappy · Oct 9, 2009

hey there,

wondering if you could help with this question:

please see attached

thank you!

untouchablecuz · Oct 9, 2009

now back to study (y)

PS: only did the essential parts

Trebla · Oct 9, 2009

I think there should an equality case in that formula....?

$$For $ n=1,\\LHS = x_1\\RHS = x_1 + 1 - 1=x_1\\LHS \geq RHS \Rightarrow $ statement true for n = 1$\\\\$Assume the statement is true for $n=k\\x_1x_2x_3....x_k \geq x_1+x_2+x_3+....+x_k + 1 - k\\\\$Required to prove the statement is true for $n=k+1\\x_1x_2x_3....x_{k+1} \geq x_1+x_2+x_3+....+x_{k+1} - k\\\\LHS = x_1x_2x_3....x_{k+1}\\\geq (x_1+x_2+x_3+....+x_k + 1 - k)x_{k+1}$ (by assumption) $\\\\=(x_1+x_2+x_3+....+x_k)x_{k+1} + x_{k+1} - kx_{k+1}\\\\\geq (x_1+x_2+x_3+....+x_k)(1) + x_{k+1} - k(1) $ as $ x_{k+1}> 1\\\\= x_1+x_2+x_3+....+x_{k+1} - k\\\\= RHS\\\\$If $n = k$ holds then $n = k + 1$ holds, but since $n = 1$ is true, the statement is true for all positive integers $n$

Affinity · Oct 11, 2009

induction with k parameters (1 Viewer)

bobohappy

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untouchablecuz

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Trebla

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Affinity

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