
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion 
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August 26th, 2016, 10:45 PM  #1 
Newbie Joined: Aug 2016 From: India Posts: 1 Thanks: 0  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)^(101) = 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)? 
August 27th, 2016, 02:07 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,756 Thanks: 696 
Are you allowed to use a calculator? I got 5.15978.

August 27th, 2016, 08:07 PM  #3  
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Quote:
That doesn't answer the question as to whether there are any shortcuts to guesstimate the answer. . Last edited by Math Message Board tutor; August 27th, 2016 at 08:10 PM.  
August 28th, 2016, 01:48 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,628 Thanks: 2077  
August 28th, 2016, 02:28 AM  #5 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 853 Thanks: 311 Math Focus: Stochastic processes, statistical inference, data mining, computational linguistics 
Use the binomial series: $\displaystyle (1 + 0.2)^9 = 1 + 9 \cdot 0.2 + \frac{9(91)}{2!} \cdot 0.2^2 + \frac{9(91)(92)}{3!} \cdot 0.2^3 + ...$ 
August 28th, 2016, 09:38 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,628 Thanks: 2077 
As 12^9 = 1728^3 = 5159780352, 10 × 1.2^9 = 51.59780352.

August 28th, 2016, 01:03 PM  #7  
Banned Camp Joined: Jun 2014 From: Earth Posts: 945 Thanks: 191  Quote:
A shortcut to guesstimating here won't be practical if someone has to do nearly as many calculations, percentagewise, as your original problem. For instance, with skipjack's method, though it is exact, first you have to cube a twodigit number. Then you have to cube a fourdigit number, which would be two successively longer operations, if done with traditional longmultiplication. 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? . Last edited by Math Message Board tutor; August 28th, 2016 at 01:07 PM.  
August 29th, 2016, 06:57 AM  #8  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
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 ballpark 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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