Manhattan Gmat Algebra Strategy Guide Book, Exponential Grow

[ozlem.ozdemir88]

Hi,

on pg 154 there is a subject named Exponential Growth under Uncommon Function Types. I did not understand either the sample question written there or the subject itself.

On pg 154 the similarity between the formula for exponential growth and the formula for an exponential sequence is mentione. However I do not know what an exponential sequence is since It has not been mentioned in any of the straregy guides. Could you pls tell in which strategy guide It explains exponential seuence?

At the bottom of pg 154 It also says "for more exponential growth see the Rates&Work Chapter of the Manhattan Guide Word Problem". However I have reviewed that too and found nothing about exponential growth..

PS: Below you can find the question:

A quantity increases in a manner such that the ratio of its values in any two consecutive years is constant. Is the quantity doubles every 6 years, by what factor does it increase in two years?

Thanks

[RonPurewal]

well, i don't have ready access to the book at the moment, but here's the basic deal: in the kind of sequence they're talking about, you always multiply each term by the same number to produce the following term.

e.g., in the sequence 2, 6, 18, 54, 162, 486, ..., you multiply by 3 at every step.

incidentally, i've never heard the term "exponential sequence" for a sequence like this one (although the term makes sense-- for instance, the terms of the sequence above have the form 2 • 3^n)

the only term i've heard for it is "geometric sequence".

so, that's what they are talking about, as far as the sequence is concerned.

[RonPurewal]

okay, now for the cool part.

if you look at a sequence like the one above, then, if you subdivide the sequence further, you get another sequence of the same kind.

since that statement clearly makes no sense without an example, here's an example.

let's take the sequence above, and stick a timeframe on it. e.g., let's say that those numbers are measurements taken every 2 hours. (measurements of what? i don't know. doesn't matter here.)
2 at midnight
6 at 2:00 a.m.
18 at 4:00 a.m.
54 at 6:00 a.m.
162 at 8:00 a.m.
486 at 10:00 a.m.

ok. but what if i want to know what happens every hour?

well, the cool part is that you'd still be multiplying by the same number each time.
so, let's say we want an hourly sequence, and that "r" is the number by which we multilply every hour. (the letter "r" is traditionally used, because it starts for "ratio of consecutive terms").
so:
2 at midnight
multiply by r --> 2•r at 1:00 a.m.
multiply by r again --> 2•r•r = 2(r^2) at 2:00 a.m.
but this is 6, so 2(r^2) = 6. thus r = √3.

now try running the whole thing, multiplying by r = √3 every hour:
2 at midnight
2√3 at 1 a.m.
2√3√3 = 6 at 2 a.m.
6√3 at 3 a.m.
6√3√3 = 18 at 4 a.m.
etc.
it "checks" with the old sequence at every step.

[RonPurewal]

now, YOU try it-- this time with the question you originally posted. (i ignored the original question on purpose, so that you get a genuine chance to try it yourself.)

if it helps, just make up specific values:
1 at time zero
2 after 6 years
4 after 12 years
etc.

now, let "r" be the multiplier every 2 years, and follow the steps i used above. (note: be careful, this time there are 3 steps between the existing numbers, rather than 2 as in my example.)