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August 26th, 2016, 10:45 PM   #1
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Needed - Trick to compute exponentials

Hi guys,

I am looking to quickly calculate the Nth term of a Geometric Progression and for that need some help with how to calculate / guesstimate exponentials

Eg. first term = 10
R = 1.2 (20% increase every year)
To find the 10th term in the series

I will use: 10 X (1.2)^(10-1) = 10 X (1.2)^9

So the question boils down to guesstimate (1.2)^9 .
I am looking at logs/antilogs but are there any shortcuts to guesstimate these exponentials (within +-10% range)?
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August 27th, 2016, 02:07 PM   #2
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Are you allowed to use a calculator? I got 5.15978.
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August 27th, 2016, 08:07 PM   #3
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Quote:
Originally Posted by techiee
but are there any shortcuts to guesstimate these exponentials (within +-10% range)?
Quote:
Originally Posted by mathman View Post
Are you allowed to use a calculator? I got 5.15978.
[
That doesn't answer the question as to whether there are any shortcuts to guesstimate the answer.



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Last edited by Math Message Board tutor; August 27th, 2016 at 08:10 PM.
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August 28th, 2016, 01:48 AM   #4
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Quote:
Originally Posted by techiee View Post
. . . are there any shortcuts . . .
1.2^9 is already short.
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August 28th, 2016, 02:28 AM   #5
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Use the binomial series:

$\displaystyle (1 + 0.2)^9 = 1 + 9 \cdot 0.2 + \frac{9(9-1)}{2!} \cdot 0.2^2 + \frac{9(9-1)(9-2)}{3!} \cdot 0.2^3 + ...$
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August 28th, 2016, 09:38 AM   #6
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As 12^9 = 1728^3 = 5159780352, 10 × 1.2^9 = 51.59780352.
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August 28th, 2016, 01:03 PM   #7
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Quote:
Originally Posted by techiee View Post
Hi guys,

I am looking to > > > quickly calculate < < < the Nth term of a Geometric Progression and for that need some
help with how to calculate / guesstimate exponentials

Then you had better use a calculator. That's the purpose to "quickly calculate."

... boils down to guesstimate (1.2)^9 .

I am looking at logs/antilogs but are there any shortcuts to guesstimate these exponentials (within +-10% range)?
I would state here that "quickly calculating" and getting a shortcut to guesstimate," *in this specific problem's context*, are contradictory/competing ideas.

A shortcut to guesstimating here won't be practical if someone has to do nearly as many calculations,
percentage-wise, as your original problem. For instance, with skipjack's method, though it is exact,
first you have to cube a two-digit number. Then you have to cube a four-digit number, which would
be two successively longer operations, if done with traditional long-multiplication.

How useful would your guesstimate be for your calculations if it need only be within 10% of the true value!?

And with 123qwerty's method, how would someone know how many terms to use until they have gotten
within your set bounds of the true value? Again, would the calculations for that method turn out to be
effort/time prohibitive?


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Last edited by Math Message Board tutor; August 28th, 2016 at 01:07 PM.
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August 29th, 2016, 06:57 AM   #8
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Quote:
Originally Posted by techiee View Post
I am looking at logs/antilogs but are there any shortcuts to guesstimate these exponentials (within +-10% range)?
Aside from series approximations (Binomial or Taylor series for example), I don't think convenient estimation methods for exponentiation exist. Exponentiation tends to compound errors so that small initial errors become large errors once the exponentiation is made. As a more extreme example, you might consider a strategy where you round 1.2 to 1.0 and then perform the exponentiation... your result then becomes incorrect by (approximately) a factor of 50. Large errors for exponentiation are a problem also for numerical solution of equations (for example, in computer software) and it requires using very precise datatypes or series approximation to avoid instability/large errors.

So... this means you're left with:

1. Use a calculator (which either brute forces the floating point arithmetic or uses a series approximation in the case of $\displaystyle e^x$)
2. Use a series approximation
3. Try to reduce the power by evaluating it explicitly in part and then (perhaps) approximate towards the final part of the calculation (a bit like SkipJack's second post)

If you're in an exam and you want to quickly check whether an exponential solution is the right ball-park without a calculator... you're out of luck. Brute force is probably your best bet. Learning powers of 2 on the top of your head can help sometimes (2, 4, 8, 16, 32, 64, 128, etc.), especially if you're doing computing, but even that's not too useful because your base is not always going to be 2.
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