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My teacher gave me a problem. Let \(1+4+10+20+35+..........+n=an^3-bn^2+cn-d\). If \(\frac{b}{a}+ \frac{d}{c}= \frac{e}{f}\) and \(gcd(e,f)=1\). What is the value of \(gcd(e+f,171)\)?

Note by Fatin Farhan
3 years, 9 months ago

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My answer is gcd(e + f, 171) = 1.

Solution: First, we must know that 1, 4, 10, 20, 35, ... form a kind of sequence. How do I know this? The sequence has a common difference in the third iteration or third operation of finding their common difference. (Sorry, I don't know how to type in Latex so I have to save time typing instead of typing and adjusting the texts). This sequence is ruled by the polynomial P(n) = (1/6) n^3 + (1/2) n^2 + (1/3) n.

Now to attack the question, since we have a sum of this sequence, therefore we have to sum up P(n). (Sorry again for cheating by using the formulas, I don't have much knowledge and mastery of Abel's Formula for Derivation.)

Since summation of n = (1/2)(n)(n+1) Summation of n^2 = (1/6)(n)(n+1)(2n+1) Summation of n^3 = ((1/2)(n)(n+1))^2

To simplify this up, Summation of P(n) = (1/24)(n^4 + 6 n^3 + 11 n^2 + 6 n) Equating this to an^3-bn^2+cn-d, we have to make equivalences since the summation of P(n) has degree of 4 while the given has 3.

This will yield: a = n/24 b = -n/4 c = 11n/24 d = -n/4 And checking these values will be equivalent to P(n).

Now to evaluate b/a+d/c=e/f (-n/4)(24/n) + (-n/4)(24/11n) = e/f e/f = -72/11 gcd (-72, 11) = 1 So e + f = -61 or 61 (This depends on where you will assign the negative sign but it does not have an effect on the gcd of these numbers.)

Final Answer: gcd(-61, 171) = gcd(61, 171) = 1 John Ashley Capellan · 3 years, 9 months ago

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@John Ashley Capellan Further information regarding my solution: 1. The common difference that I am referring to is that when you get the differences of each terms in the sequence. In the first operation, the differences (for the first 5 terms) are 3, 6, 10, and 15. In the second operation, we obtain differences of 3, 4, and 5. And finally, in the third operation, we get the uniform difference of 1.

  1. The summations are governed by the rule of summation wherein the summation of sums, we can distribute their summations each term. And the other rule such that the summation of a product of a constant and a variable is equivalent to the constant times the summation of the variable.
John Ashley Capellan · 3 years, 9 months ago

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