Can an Endless String of Numbers have a Sum?

If you tutor math 12, you’ll have come across the premise

1 + 1/2 + 1/4 + 1/8 +….

Before we go any further, a reminder on how to multiply fractions:

(2/3) x (4/5) = 8/15.  You just multiply the two top numbers and put that number on top.  Then you multiply the two bottom ones and put that number on the bottom.

Now, back to 1 + 1/2 + 1/4 + 1/8 + 1/16+….(the terms continue forever)

To generate the next term, you just multiply the previous one by, in this case, 1/2.

(1/4) x (1/2) = 1/8

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 +….

In spite of the infinite number of terms, the answer is 2.

1 + 1/2 + 1/4 + 1/8 + 1/16 +1/32 + 1/64 + (the terms go on forever)….= 2.

How can there be a sum of an unending number of terms?  Well, there is a proof for it – but that doesn’t mean you have to believe it.  Perhaps the best answer is that from a mathematical point of view, you can add up an infinite number of terms.  However, in this context, two things have to be true:

1) You must always multiply the current term by the same factor to get the next one.

2) That multiplying factor must be less than 1.

So there isn’t a sum to 1+1/2+1/3+1/4 + (the terms continue forever)….because the next number is not a constant multiple of the one before.

There isn’t a sum to 1 + 1 + 1 + (the terms continue forever)….because we’re multiplying the current term by exactly 1 to get the next term.  We must multiply by the same number each time, but it must also be less than one.

There is a sum to 1 + 2/3 + 4/9 + 8/27 +….(infinite terms).  Note that we are always multiplying by 2/3 to get the next term.  The sum is 3.

The formula for the sum is

S=a/(1-r), where

a is the first term

r is the multiplying factor to get the next term (called the common ratio)

If you use a calculator, be sure to put the brackets around the denominator as shown in the formula.

Thanks for stopping by.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Julius Caesar: Cassius, V,iii

Caesar, thou art avenged, even with the sword that kill’d thee.

Cassius, Act V, scene iii, Julius Caesar

In Act V, scene iii, of Julius Caesar, Cassius commits suicide.  Having been misled that his dear friend Titinius is slain, Cassius decides he must join him.

For almost two years, I wondered why Cassius had decided to die at that moment.  He may have loved Titinius, but Cassius was a soldier; he’d surely lost dear friends before.  I read Julius Caesar in grade ten and loved it.  By December of grade twelve, though, I was still asking myself why Cassius gave up when he did.

However, I had a resource in December, 1987, that I’d lacked when I’d first read Julius Caesar:  a waiter with whom I worked, named Tim.  I assisted him some nights at a pretty fancy place.  One night, Julius Caesar just happened to come up somehow, so I asked Tim why Cassius had thrown in the towel so early in the battle.

“Simple:  guilt,” Tim replied.  “Cassius knew he shouldn’t have led the conspiracy to kill Caesar.  He regretted it.  The guilt was too much for him.”

At once, the question was resolved.  It’s remained so since December, 1987.  Of all the answers I’ve received my whole life, that one from Tim remains possibly the most satisfying.

Thanks for stopping by.  Please come again.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

The Semicolon

Welcome back.

While tutoring English, punctuation is a constant concern.  Using a variety of punctuation leads to sentence variety – something that always impresses English teachers.

The semicolon can be used in many ways, but its simplest use is to join two short, but complete sentences.

a)  The weather turned to rain.  They canceled the game.

can become

b)  The weather turned to rain; they canceled the game.

The construction in b) brings the writing to a much higher level.  Note that even though the semicolon joins two complete sentences, the resulting sentence can still be pleasantly short and inviting to read.

Don’t overuse semicolons.  A few times a page is enough.

Happy writing. See you next time.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Divisibility by 3, Divisibility by 9

During math tutoring, no matter what the level, the front line is arithmetic.

In that spirit, here’s a handy trick to tell if a number is divisible by 3:

Add up the digits.  If the sum is divisible by 3, then so is the original number.

Example:  consider the number 222.  2+2+2=6.  Since 6 is divisible by 3, so is 222.  (Many people might not expect that.)  In fact, 222 ÷ 3 = 74.  (Check for yourself.)

A similar law works for 9:  if the digits add up to something divisible by 9, then the original number is also divisible by 9.

Example:  consider 414.  4+1+4 = 9, so 414 is divisible by 9.  So is 864, since 8+6+4=18, and 18 is of course divisible by 9.

Of all the math tricks I know, these are two of the most useful.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

French: passé composé, imparfait, and plus-que-parfait: when to use them.


Well, today I thought we’d discuss some French:  specifically, the passé composé, the imparfait, and the plus-que-parfait.  In what situation do you use each?

The passé composé is the first way to express the past tense that I learned in high school.  It is the French equivalent to the English verb with -ed (eg., I walked).  The passé composé has two parts:  the auxiliary, followed by the past participle.

The imparfait expresses, as my French teachers always explained, “a state of things that had no particular beginning.  It may not yet be finished.”  For instance, “When I was young ….”  Additionally, “It was raining….”  Both use the imparfait tense.  In French, it’s among the easiest constructions, consisting of a stem with a subject-specific ending.

The plus-que-parfait expresses a completed action that happened before another completed action.  Consider  the sentence:  “I had finished the laundry when you called.”  “I had finished” is the plus-que-parfait tense, whereas “you called” is the passé composé.

Well, there you have it:  the passé composé, the imparfait, and the plus-que-parfait.  My wife (who is French and does our French tutoring) explained them to me.

Have a great day.  Come again.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Simplifying Radicals…part I

Hello.  One of the most dreaded topics in high school math – as you’ll quickly find out if you do math tutoring – is radicals.

Well, here’s our first math post: simplifying radicals. Today, we’ll look at square roots only.

For example, consider the following:

Simplify √90

Well, here’s what we do:

1)  Find the number that divides into 90 and is “square rootable”.

In this case it’s 9, since √9 is exactly 3.

2)  Realize that √90 =  (√9)(√10)

Since 9×10=90.

3)  Replace √9 with 3.

(√9)(√10) = 3√10

So √90 = 3√10.

This is a basic case of simplifying a radical.

Come back for more hints.  Have a great day!

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Testing, testing…

Good afternoon.  This is the first blog entry by Oracle Tutoring, Campbell River, BC.  We tutor math, chemistry, physics, biology 12, English, French, and science.

We will be doing posts that help with specific problems or queries in the subjects above.

Thanks for coming.  Please visit again soon!