5?

I am sure the statement is true for Flame Haze! As for me, I wish I could spend more time on it whilst still at university.

Ah well, this is why I am here! To polish the math/thinking skills!

I think it is correct but we shall see what the others have to say about this! Well done!

The question can be solved algebraically using knowledge learned from SPM Progressions, Indices and Logarithms.



which can be rearranged to give

.

By letting

,

we need find integer n which satisfies



Using a graphing software, one can evaluate

Hi Flame Haze, RED-HAIR-SHANKS, maximR, crazywing26 and studyboy,

The purpose of these exercises is to train your problem-solving skill to be adaptive to searching viable mathematical methods from 360°, primarily Arithmetic, Algebra, Polynomials, Graphs, and Linear equations. It does not limit to the formulas you learned in Sequences & Series.

After simplification, the value of



is a proper fraction in its lowest form, which is related to Gauss' famous number. Find the difference of its denominator and numerator.

A standard way to solve this type of problem in STPM, taking the logarithm at both sides:



Hence, the least value of n is 5.

This post has been edited by ystiang: Dec 25 2013, 05:39 PM

Hi it's a chemistry question here?? Which bond.is.stronger?? Ionic bond.or.ionic bond.with covalent characterics??

Fantastic solution! Normally one wouldn't think of using graphs for this sort of stuff!

This might be unknown to some of you here but such a method is commonplace in academia. A graph provides much insight on a particular equation in terms of behaviour, possible roots and etc. Very powerful stuff!

This post has been edited by studyboy: Dec 25 2013, 05:58 PM

1/5050, 5049?

This post has been edited by Flame Haze: Dec 25 2013, 07:07 PM

I think you are correct!

However, how would one prove this by induction? Did you do it by induction Flame Haze?

Critical_Fallacy or anyone else do comment!

I did it using similar method as the last question.

Thanks for the information!

The answer is 5, yes.
ystiang provides the standard answer as according to the answer scheme, but it's worth noting that there is more than one method that can be applied, as shown by Critical_Fallacy. Depending on the seniority of the teacher marking your paper, the alternative method may/may not be accepted (younger teachers tend to be very rigid when it comes to marking papers according to the answer scheme given). The only time that an answer is to strictly follow a specific method is when you see the word "hence" in the question, which implies that you MUST use the results from the previous section (either an equality or a graph) to deduce your answer.

Flame Haze-san was able to grasp the nature of series and rational functions. With enough training, one should be able to see this pattern:



Since it is a rational function, you can do a partial fraction decomposition!

I remember I did this kind of question in my SPM. And v1n0d is right. Under normal exam circumstances, one may not be able to plot a smooth curve for evaluation because graphing calculators are usually not allowed. Technically, it is possible to sketch by computing n = 1, 2, 3, 4, ... and then link up the dots. But it is impractical because one does not know the root, Depending on the convergence of the series, the root can occur at n = 25 or at n = 50. Usually the questions are single-variable problems and most of them have closed-form solutions.

My purpose of showing the graphical solution (without spilling out the algebraic solution ), as mentioned by studyboy, is to create the awareness in students that we can model a function and analyze it better by understanding the behaviors of the function f(x) at -∞ < x < ∞ domain. In analyzing dynamic systems in Biology, Chemistry, Physics, the graphs tell us many useful information for design purposes, such as the system response (shape), resiliency (boundedness), stability (convergence), optimality (max performance / min resources), etc.

What happen to the "dots"? I don't have a textbook with me. What kind of formations are written in your Chemistry textbook about the Ionic Bonding and Covalent Bonding? Perhaps you can tell me!

Hi Flame Haze, RED-HAIR-SHANKS, maximR, crazywing26 and studyboy,

I've updated Post #1 with Calculators.

Scientific Calculator


Graphing Calculator


Programmable Calculator


Equations and Systems Solver


Symbolic Differentiation and Integration

Hi Critical_Fallacy and crazywing26, I would like to ask you something concerning matrices.

Suppose that we have a 3 x 3 matrix A, and that we have to find it's determinant.


From the above matrix A, it's determinant can be evaluated by adding the products of any elements with any one row(with it's column) with their respective cofactors. It should be like this:


But, I too found out that there is another alternative way to solve the determinant of a 3 x 3 matrix, that is by using Sarrus' Rule. It's the sum of the multiplication of entries in every arrow to the right(blue colour) minus the sum of multiplication of entries in every arrow to the left(red colour).


Now my question is, as far as I know from my book, Sarrus' Rule is not mentioned in it. If by any chances were I to stumble across a question concerning the determinant of a 3 x 3 matrix, can I use Sarrus' Rule? My reason is that it's way more flexible and less tedious compared to the other method in which we need to first find the minor and cofactor first. But, knowing that sometimes the marking scheme for STPM is extremely stringent and rigid, I feel that it's not a safe bet. Any Form 6-ers have experience this before?

Another thing I would like to know is why the signs of the 3 x 3 matrix (cofactors) should be like this ? :


This post has been edited by RED-HAIR-SHANKS: Dec 26 2013, 06:46 PM

the sign is like that because the formula states that (-1)^i+j (M)

try finding books on linear algebra, its really fun to read

http://en.wikipedia.org/wiki/Leibniz_formu...or_determinants

http://en.wikipedia.org/wiki/Laplace_expansion

http://en.wikipedia.org/wiki/Cofactor_%28linear_algebra%29

Have a look at these links!

As for the Rule of Sarrus, use it by all means to check your answers. To be on the safe side, it is better if you use the standard way of computing a 3X3 matrix and by standard, I mean whatever that you are taught in Form 6.

This post has been edited by studyboy: Dec 26 2013, 07:10 PM