How do I get an expression to verify by induction?
#mathematical induction
tawny sedge
coarse hamletBOT
How do I get an expression to verify by induction?
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tawny sedge
But how can I rewrite sum from 1 to n+1 as sum from 1 to n + something
timid remnant
$$ \sum _{k=1}^{n+1} \frac{1}{k+n} \geqslant \frac{7}{12} + \frac{1}{2n+1} $$
tender vergeBOT
aL
timid remnant
does this look right?
tawny sedge
Can you explain how you get 1/(2n+1) ?
timid remnant
put k=n+1, what do you get
tawny sedge
Makes sense
timid remnant
what do you mean solve it?
this inequality is already enough to prove induction step
tawny sedge
I see it makes sense but my teacher gave this tip so I was trying to get this
timid remnant
i don't see why subtraction is even necessary here
I get that you have to use induction per the exercise instruction, but actually you dont even need it
when you calculated base case
the proof is already done
from there you only ever add positive numbers on top
if you want to formulate it by induction then it is enough to write this
$$\sum _{k=1}^{n+1} \frac{1}{k+n} \geqslant \frac{7}{12} + \frac{1}{2n+1} > \frac{7}{12} $$
tender vergeBOT
aL
timid remnant
the claim is true for n+1 and you're done
tawny sedge
the proof is already done
I see interesting point
I bet My teacher would like me to say that on the exam he is always saying to solve it the simplest way and if I remember correctly he already showed a look alike exercise where we just prove the 1st term for a fixed RHS
tawny sedge
I see it makes sense but my teacher gave this tip so I was trying to get this
I was just curious how to get to this I don’t see where the subtraction comes from
timid remnant
I see it makes sense but my teacher gave this tip so I was trying to get this
this is true though
autumn fjord
tender vergeBOT
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willow spade
because $a^2_n=n+\sqrt{(n-1)+\sqrt{(n-2)+...\sqrt{2+\sqrt{1}}}$
and $a^2_n-n=\sqrt{(n-1)+\sqrt{(n-2)+...\sqrt{2+\sqrt{1}}}$
tender vergeBOT
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long otterBOT
@tawny sedge
swift sluice
+close