1) sum of series = (n*(n+1))/2
eg from 1 to 19,
sum = (19*20)/2 = 190

The “diagram” below shows the reasoning behind this for the series 1,2,3,4. The ‘*’ represents the value of numbers 1,2,3 and 4. To find the number of *s, add an equal number of ‘0’s to form a rectangle. This will form a 4 x 5 rectangle. The number of *s is given by (4*5)/2

12345
1 *0000
2 **000
3 ***00
4 ****0

2) sum = ((1+n)/2) * n
sum = ( (1+19)/2 ) * 19 = 190

This finds the average of the series (1+n)/2 and multiply the average by the number of terms in the series. Average of 1 to 19 is (1+19)/2 or 10. Multiply by 19 to get the sum of 190.

3) Multiply the median number of the series by the number of terms in the series. This is somewhat similar to (2) above.

If n is odd, the median number is (n+1)/2. The median number for 1 to 19 is (19+1)/2 or 10. Sum of 1 to 19 is 10 * 19 = 190.

If n is even (eg 1 to 20), there is no exact median number. Therefore to get the sum of 1 to 20, first get the sum of 1 to 19 (as above) then add the last number 20 to get the final sum. sum of 1 to 20 is 190 + 20 = 210

Right on the money. This is also the way that is described on this Mathmojo page about triangular numbers.

I tell my students this is just like counting the number of people in a crowd by counting the total number of legs and dividing by two.

I couldn’t agree more. Yes, it’s not “politically correct,” but it’s right. Although I wouldn’t always knock community college grads. A lot of hard-working people have had to put themselves through school while keeping a job, and it could be a stepping-stone to greater heights. OK, yeah, it’s often just a waste of time, too.

Maybe the schools won’t even exist much longer, seeing how they and the economy are going. Maybe that’s not a bad thing. More people will have to take their own initiative. It could be a blessing in disguise.

At any rate, it seems like you are on the right path if you have been googling around to improve your math skills. Thanks for your comments, and power to you!

This never has happened because through grade school all my math teachers always complained about the school administration, bad wages, enforcing the pointless DRESS CODE, and would often say “but you don’t need to know that, so we will just skip it…” For so long I have felt guilt, blaming my parents’ genes even, for my lack of skills in math..

If I could go back in time, I would…and I would choose a school district that did NOT have superintendents without Ph.Ds, weren’t the gold-digging wives of doctors, board members weren’t bankers/random business owners, and the principals weren’t community college material.. Excuse my language, but grade school really did screw me over, and many of my peers. Had I been exposed to a good education that made me apt in math, i dont think i would be studying foreign affairs, i would likely be studying aerospace engineering… REFORM THE EDUCATION SYSTEM! IT’S SCREWING OVER OUR GENERATION…

Start by looking at the facts, and a whole new world of understanding may open for you. You won’t need to defend inadequate beliefs anymore.

Please read the posts I linked to above. They give cogent reasons that why division by zero renders what it does and doesn’t render.

If you need a more informed and very thorough article, when you check out the “Why We Don’t Divide By Zero in Arithmetic” link, go to the bottom and click on the link for “The Zero Saga.” It is an education in itself.

To be very specific, on reason that your reason won’t stand rationally, is that when you divide something by, say, seven, you don’t expect it to end up having seven parts. It may have any number of parts. When you divide 28/7 you get 4 parts, not seven parts. So we wouldn’t expect dividing something by zero to give you zero parts in the first place. We would expect a different answer for every dividend.

You can start understanding the real reason we don’t divide by zero in arithmetic by using that as a clue.

Happy learning!

- Professor Homunculus

Amen, brother! I was getting a little frustrated with the plethora of opinion and lack of input.

To your question: I really don’t know if it’s about the letters or the context. That’s a great point. There can be many, many ways to think about this.

I always feel a little cheated by “IQ” questions like this. They present it as though there is some “best” answer, but nowhere do they provide it to check your own answer against, nor any reasoning they might have for it.

On the other hand, I love open-ended questions that make you think. So I take this one as open-ended, and I seriously doubt that there is an authoritative solution.

Actually, I think there cannot be one. The parameters have not been set.

One thing I like about math is that, in general, the parameters have been very clearly establishes (to the best of our human abilities, so far, at any rate.). The axioms are clear, and everything follows from them (without getting into Goedel).

In base 10, 2+2=4, all other things being equal.

But is @ + & = 2, 5, or H? Which answer is best? It’s a nonsense question unless some criteria for an answer are given. We can contemplate it all day. It’s like a Zen koan.

“What is the sound of one hand clapping? I want the best answer, now, or you fail!” Nah, Zen doesn’t work that way (does it?)

Your approach is refreshing. Who says it has to be about the letters? I was even thinking that “dough” has four letters that have the sound of one vowel, so maybe all four letters are considered vowels in this case. Maybe that could have something to do with it. The sky’s the limit.

The only thing that can stop us from thinking is when we think we have the final answer. Then we have come to the end of thought. How sad.

Thanks so much for your comment, Dock.