It's your preference. To answer the rest, please concentrate on reading my post CLOSELY.

As mentioned in previous post. There is a mistake in (1), which is affecting (3) and (4).

(1) Find the electric potential at the point β (far right) due to a positively charged particle , separated by a distance .

You can substitute the positively charged particle = 5q and the negatively charged particle = -3q, as in your original problem in Post #396.

This post has been edited by Critical_Fallacy: Dec 24 2013, 01:18 AM

Doesn't this figure tell you the whole story clearly?

When solving physics problem, you cannot rush. Read carefully and understand what are required by the questions. They are your mini tasks, sub-problems that you need to tackle.

Easy is GOOD! Try the next one.

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

What a year it's been. This is 2013 end-of-year question.

Evaluate .

P.S. If you can tackle the previous problem, you can tackle this one too.

You have a sharp mind!

Very close!



BINGO! But I've given you a gentle clue! Remember the last series?





This post has been edited by Critical_Fallacy: Dec 24 2013, 01:38 PM

This is the full solution by applying the special factorization :

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

OK! Here is the real deal. This is STPM/A-level question, but if you had been following my tutorial faithfully from the beginning, or if you passed your 1st-term Pure Math exam with flying colors, this one is probably peanut to you!

Find the sum .

So, my hypothesis on passing the Pure Math exam with flying colors has a good degree of trueness.

The proper way of using the sigma notation in this question is probably:



In real exam, depending on the marks allocation, Flame Haze probably lose ½ to 1 mark.

This question is in fact, a hybrid of series and partial fraction to test your skills.

Polynomial 101:

(1) For a rational function (two polynomial functions in quotient form), if the numerator is larger than the denominator, you can perform a polynomial long division (or a compact method called Synthetic Division).

(2) If the denominator of the remainder from the division can be factorized into irreducible polynomials, you can do a partial fraction decomposition on the remainder.

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.

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

The diagonal scheme of expanding the determinants or “Sarrus' rule” is generally CORRECT ONLY for determinants of second- (2×2 matrix) and third-orders (3×3 matrix). For determinants of higher order, you must pay attention to the formal definition of determinants.

In discussing a general nth order determinant, it is convenient to use the double-subscript notation. The determinant itself is denoted by a variety of symbols. The following notations are all equivalent:



The value of a nth order determinant is given by this formal equation:



where is the Levi-Civita symbol. The properties characterized by the Levi-Civita symbol are defined as follows:



An easy way to determine whether a given permutation is even or odd is to write out the normal order and write the permutation directly below it. Then connect corresponding numbers in these two arrangements with line segments, and count the number of intersections between pairs of these lines. If the number of intersections is even, then the given permutation is even. If the number of intersections is odd, then the permutation is odd. For example, to find the permutation (2, 3, 4, 1), we write out the normal order and permutation in the diagram.



There are three intersections. Therefore the permutation is odd and the Levi-Civita value is −1. Let's do an example to find the determinant of a 3×3 matrix.



This post has been edited by Critical_Fallacy: Dec 26 2013, 10:40 PM

If you are worried on the marking scheme, here is a surefire way to compute the determinant of a 3×3 matrix:

Let . Find det(A) by cofactor expansion.

SOLUTION:

Step 1: Pick a row or a column which contains 0's in the entries. In this case, I'd pick either Row 1 or Column 3 because a13 = 0. Say we pick Row 1, then:



Step 2: Compute the determinant using the built-in [MAT] function in your scientific calculator CASIO fx-570, and then write the value:

Finding the determinant of a 4×4 matrix is still manageable:



Now, using CASIO fx-570, compute the determinants of the 4 cofactors:



Finding the determinant of a true 5×5 matrix without 0 entries is a real painstakingly tedious job. There are FIVE 4×4 cofactors. To find the determinant of each cofactor, you have to compute FOUR determinants of the 3×3 minor cofactors (see above example). Therefore, FIVE (5) × FOUR (4) = TWENTY (20) computations!