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What is two-third of the cube of a number?
2/3n3
What is the next term in this sequence 7 12 30 70 141 252?
Given any number, it is always possible to find a polynomial of degree 6 that will fit the above numbers and the additional given number.The simplest position to value rule, in polynomial form, for the above sequence isUn = (3n3 - 5n2 + 4n - 12)/2 for n = 1, 2, 3, ...and accordingly, U7 = 412.
What is the pattern in one eighth two sevenths one half and four fifths?
One possible answer is: Un = (3n3 - 3n2 + 78n - 8)/560 for n = 1, 2, 3, 4.
What is the next number in this pattern 3 12 35 63?
Given ANY number, it is easy to find a polynomial of order 4 such that if you use it as a position to value rule you get the four given numbers and your chosen one as the fifth. As a result, any number can be the next in the sequence. The simplest polynomial of order 3 for the above four numbers is: Un = (-3n3 + 32n2 - 57n + 34)/2 for n = 1, 2, 3, ... Accordingly, the next number is 87.
The product of four consective integers is one less than a perfect square?
Suppose the smallest of the integers is n. Then the product of the four consecutive integers is n*(n+1)*(n+2)*(n+3) =(n2+3n)(n2+3n+2) = n4+6n3+11n2+6n So product +1 = n4+6n3+11n2+6n+1 which can be factorised as follows: n4+3n3+n2 +3n3+9n2+3n + n2+3n+1 =[n2+3n+1]2 Thus, one more that the product of four consecutive integers is a perfect square.
