Re: Mathematical Induction

Hello, BACONATOR!

Quote:

$\displaystyle \text{18. Consider the following four equations:}$

$\displaystyle \begin{array}{cccccc}(1) & 1 &=& 0^3 + 1^3 \\(2) & 2+3+4 &=& 1^3+2^3 \\ (3) & 5+6+7+8+9 &=& 2^3+3^3 \\ (4) & 10+11+12+13+14+15+16 &=& 3^3+4^3 \end{array}$

$\displaystyle \text{Conjecture the general formula suggested by these four equations,}$

. . $\displaystyle \text{and prove your conjecture.}$

The left side is an arithmetic series.

. . Its first term is: $\displaystyle a = k^2 - 2k+2$

. . The common difference is: $\displaystyle d = 1$,

. . The number of terms is: $\displaystyle n \,=\,2k-1$

The sum is: .$\displaystyle S \;=\;\frac{n}{2}\big[2a + (n-1)d\big]$

. . . . . . . . . $\displaystyle S \;=\;\frac{2k-1}{2}\big[2(k^2-2k+2) + (2k-2)1\big]$

. . . . . . . . . $\displaystyle S \;=\;\frac{2k-1}{2}\big[2k^2 - 4k+ 4 + 2k - 2\big] $

. . . . . . . . . $\displaystyle S \;=\;\frac{2k-1}{2}\big[2k^2 - 2k + 2\big]$

. . . . . . . . . $\displaystyle S \;=\;(2k-1)(k^2 - k + 1)$

. . . . . . . . . $\displaystyle S \;=\;2k^3 - 2k^2 + 2k - k^2 + k - 1$

. . . . . . . . . $\displaystyle S \;=\; 2k^3 - 3k^2 + 3k - 1$

. . . . . . . . . $\displaystyle S \;=\;(k^3 - 3k^2 + 3k - 1) + k^3$

. . . . . . . . . $\displaystyle S \;=\;(k-1)^3 + k^3$

Re: Mathematical Induction

I'm curious. You replied to your own post from 5 years ago with an almost identical one?

-Dan