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How would I express $\sum_{i=0}^n (3𝑖^3 βˆ’ 6𝑖 + 2)$ as a polynomial $p(n)$ and also prove that the sum equals $p(n)$ using induction?

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  • $\begingroup$ I don't understand the mathematical expression, you wrote. Is it like "zero times" the expression in brackets? $\endgroup$ – thinkingeye May 24 at 5:31
  • $\begingroup$ @thinkingeye It is meant to say "The sum of i=0 to n of (3𝑖^3 βˆ’ 6𝑖 + 2)". Sorry for the confusion, I have made the clarification in the original post :) $\endgroup$ – Daniel May 24 at 5:34
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    $\begingroup$ always try to include your attempt when you ask a question on this site. $\endgroup$ – Siong Thye Goh May 24 at 10:13
  • $\begingroup$ Please accept the answer, if it solved your problem. $\endgroup$ – thinkingeye May 24 at 11:38
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There are formulas for the summation of powers, to find for example here: Sum of $n$, $n^2$, or $n^3$

\begin{align} p(n) = \sum_{i=0}^n \left(3i^3 - 6i + 2\right) &= 2 + \sum_{i=1}^n \left(3i^3 - 6i + 2\right) \\ &= 2 + 3 \sum_{i=1}^n i^3 - 6 \sum_{i=1}^n i + 2 \sum_{i=1}^n 1 \\ &= 2 + 3 \frac{n^2 (n+1)^2}{4} - 6\frac{n(n+1)}{2} + 2n \\ &= \frac{3}{4} n^4 + \frac{3}{2} n^3 + \frac{9}{4} n^2 - n + 2 \end{align}

For the proof by induction, you have to show that the left hand side (LHS) equals everytime to the right hand side (RHS).

Lets assume, it is true for all $p(m)$ with $m < n$. Lets look at the step from $n-1$ to $n$.

Step difference on RHS:

$$ \begin{align} p(n) - p(n-1) &= \frac{3}{4} n^4 + \frac{3}{2} n^3 + \frac{9}{4} n^2 - n + 2 \\ &- \frac{3}{4} (n-1)^4 - \frac{3}{2} (n-1)^3 - \frac{9}{4} (n-1)^2 + (n-1) - 2 \\ &= 3 n^3 -6n + 2 \end{align} $$

Step difference on the LHS:

\begin{align} p(n) - p(n-1) &= \sum_{i=0}^n \left(3i^3 - 6i + 2\right) - \sum_{i=0}^{n-1} \left(3i^3 - 6i + 2\right) \\ &= 3n^3 - 6n + 2 \end{align}

So, the all step differences are the same on LHS and RHS.

Now, lets research the base case $p(0)$. And as you will see, they are the same on LHS and RHS, too.

So, if the start is the same and the difference at each step is the same, then each step is the same.

Ergo: LHS $=$ RHS.

\begin{align} p(n) = \sum_{i=0}^n \left(3i^3 - 6i + 2\right) &= \frac{3}{4} n^4 + \frac{3}{2} n^3 + \frac{9}{4} n^2 - n + 2 \end{align}

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