Welcome to The Puzzler. Every week, there are 2 new puzzles related to my passion: math, logic, and thinking. The first puzzle will be The Puzzler Quik, meant for those who crave something fun-sized. The second puzzle will be The Puzzler Think, meant for those who love to ponder. The answers will be posted in next week’s column. Don’t forget to submit your answer for a potential shout-out in the next edition of the Puzzler.
Puzzler Quik
Can you simplify this expression? $$\log_{23}{23} + \log_{23}{1} + \log_{16}{256}?$$
Puzzled?
Don’t know what Logarithms are? Tune out for our upcoming video on them!
Try simplifying each log seperately!
What is the value of 
for which 
?
Puzzled?
Don’t know what Logarithms are? Tune out for our upcoming video on them!
Try making the base common in all 3 logarithms!
ANSWERS TO LAST WEEK’S PUZZLERS
Last Week's Puzzler Quik
Anay from Portland, Oregon was the winner of this shout-out.
There is a formula for this! $$\frac{n(n+1)(2n+1)}{6}$$ Plugging in 100 for “n”, we get $$\frac{100\cdot101\cdot201}{6} = 338350.$$
Our answer is 338350!!!
Want a proof for this formula? Here is a good link! https://proofwiki.org/wiki/Sum_of_Sequence_of_Squares
Last Week's Puzzler Think
This week’s shout out goes to Chirag from Arkansas! Kudos to him!
Here is his awesome explanation:
It is well-known that the sum of an infinite geometric series is $$\frac{a}{1-r}$$
But nothing is good without a proof. So here is one.
Let $$a + ar + ar^2 + … = S.$$ Thus, $$ar + ar^2 + ar^3 + … = S\cdot{r}$$ Subtracting this equation from the first equation, we have $$a = S – Sr => a = S(1 – r).$$ Dividing both sides by 1 – r, $$\frac{a}{1 – r} = S.$$ In this case, a = 1/2 and r = 1/2, so substituting it into the formula, $$S = \frac{\frac{1}{2}}{1 – \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} =1.$$ We have proved that $$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+… = 1.$$
We are done!
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