What is the NO-SHORTCUT approach for learning great Mathematics?

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Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2010 based on functional equation.

Consider the identity \(1+2+......+n = \frac {1}{2}n(n+1)\). If we set \(P_{1}(x) = \frac{1}{2}x(x+1)\) , then it is the unique polynomials such that for all positive integer n,\(p_{1}(n) = 1+2+..............+n\) . In general, for each positive integer k, there is a unique polynomial \(P_{k} (x) \) such that :

\(P_{k} (n) = 1^k + 2^ k+3^k +..................+n^k\) for each n =1,2,3...............

Find the value of \(P_{2010} (-\frac {1}{2})\) .

- 2
- 5
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Polynomials

Functional Equation

But try the problem first...

Answer : 0

Source

Suggested Reading

Singapore Mathematics Olympiad

Challenges and Thrills - Pre College Mathematics

First hint

If you got stuck in this question we definitely can start from here:

In the question given above say k is the positive even number :

so let \(f(x) = P_{k} - P_(x-1)\)

Then \(f(n) = n^k \) for all integer \(n\geq 2\) (when f is polynomials)

Like this then \(f(x) = x^k\)(again for all \( x \geq 2\) .

Second Hint

If you got stuck after first hint try this one

\(P_{k} (-n + 1) - P_{k}(-n) = f(-n +1) = (n-1)^k\)..................................(1)

Again, \(P_{k} (-n + 2) - P_{k}(-n+1) = f(-n +2) = (n-2)^k\).......................................(2)

Now taking n = 1;The \(eq^n\)(1) becomes, \(P_{k}(0) - P_{k}(-1) = f(0) = 0^{k}\),

And for \(eq^(n)\) (2) ; \(P_{k}(1) - P_{k}(0) = f(1) = 1^{k}\).

Now sum these equation and try to solve the rest...........

Final Step

Summing this two equation we get , \(P_{k}(1)-P_{k}(-n) = 1^{k} + 0^{k}+1^{k}+.......+(n-1)^k\).

so , \(P_{k}(-n)+P_{k}(n-1)=0\)

Again if \(g(x) = (P_{k}(-x)+P_{k}(x-1)\)

Then g(n) is equal to 0 for all integer \(n\geq2\)

As g is polynomial, g(x) =0;

So , \(P_{k}(-\frac {1}{2}) + P_{k}(-\frac {1}{2}) = 0\)

so \(p_{k}(-\frac {1}{2}) = 0\) ............(Answer)

- https://www.cheenta.com/power-mean-inequality-for-math-olympiad/
- https://www.youtube.com/watch?v=PfRqs9W8nPQ&t=2s

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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