More Polynomial Equations

337 /www/nrich/html/content/00/11/15plus4/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. $$\begin{eqnarray} a(n) &= &1 + 2 + 3 + ... + n \\ b(n) &= &1^2 + 2^2 + 3^2 + ... + n^2\\ c(n) &= &1^3 + 2^3 + 3^3 + ... + n^3. \end{eqnarray}$$ It is well known that $c(n) = a(n)^2$ . What are the relationships between $a(n)$ and $b(n)$ and between $b(n)$ and $c(n)$?</mdoxml> <mdoxml version="1.0"> Jim sent in this solution, using the ideas from our hints. $a(n)=1+2+\ldots+n=\frac{n(n+1)}{2}$ $b(n)=1^2+2^2+\ldots+n^2=\frac{n(n+1)(2n+1)}{6}$ $c(n)=1^3+2^3+\ldots+n^3=\frac{n^2(n+1)^2}{4}$ It's obvious that $c=a^2$, from this. Also, $2a=n^2+n$, so, solving the quadratic (and using the fact that $n> 0$), we get $n=\frac{-1+\sqrt{1+8a}}{2}$. Now substitute this for $n$ in $b$, to get $b=\frac{n(n+1)(2n+1)}{6}=\frac{a}{3}\times(2n+1)=\frac{a}{3}\times\sqrt{1+ 8a}$. So $3b=a\sqrt{1+8a}$, so $9b^2=a^2+8a^3$. Now we can combine these two expressions: $9b^2=c+8c\sqrt{c}$, so $8c\sqrt{c}= 9b^2-c$, so $64c^3=81b^4-18b^2c+c^2$. (It's easy to check that the expressions above in terms of $n$ do work in this!) </mdoxml> <mdoxml version="1.0"> </mdoxml> <mdoxml version="1.0"> Here you need to start with $$\begin{eqnarray} a(n) &= &1 + 2 + 3 + ... + n &= & {1\over 2}n(n+1)\\ b(n) &= &1^2 + 2^2 + 3^2 + ... + n^2 &= & {1\over 6}n(n+1)(2n + 1)\\ c(n) &= &1^3 + 2^3 + 3^3 + ... + n^3 &= & {1\over 4}n^2(n+1)^2. \end{eqnarray}$$ These are well known results found in many text books and you will find proofs in some problems on this website. See <a href="../public/viewer.php?obj_id=325">Picture Story</a> and <a href="../public/viewer.php?obj_id=329">Natural Sum.</a> </mdoxml> 2 4 0 0 0 0 1 More Polynomial Equations Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively. Advanced Algebra Summation of series Advanced Algebra Relations Advanced Algebra Polynomials Algebra Manipulating algebraic expressions/formulae