How to expand
$(x_1 + x_2 + x_3 + x_4 + x_5 +\cdots+x_n)^{2}$. Is their any general formula for this? Thanks
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1Have you ever heard about the multinomial formula? – Hanul Jeon Aug 16 '15 at 06:17
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@tetori No i haven't – Taylor Ted Aug 16 '15 at 06:26
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That is the generalization you might want. – Hanul Jeon Aug 16 '15 at 06:28
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Here's a link to a general solution https://en.wikipedia.org/wiki/Multinomial_theorem – user103828 Feb 02 '17 at 08:50
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$$\left( \sum_{i=1}^n x_i \right)^2 = \sum_{i=1}^n \sum_{j=1}^n x_i x_j = \sum_{i=1}^n x_i{}^2 + 2 \sum_{1 \leq i < j \leq n} x_i x_j.$$
A particular example is: $$(a+b)^2 = a^2 + 2ab + b^2.$$
wythagoras
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Yes, there is. $$(x_1+x_2+x_3+\cdots+x_n)^2=\\ x_1^2+x_1x_2+x_1x_3+\cdots+x_1x_n+ \\ x_2x_1+x_2^2+x_2x_3+\cdots+x_2x_n+ \\ x_3x_1+x_3x_2+x_3^2+\cdots+x_3x_n+ \\ \cdots \\ x_nx_1+x_nx_2+x_nx_3+\cdots+x_n^2 $$
wythagoras
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So it is just square of specific term and its product with other terms – Taylor Ted Aug 16 '15 at 06:32
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Yes, basically. When there are say three terms, I usually write the squares before the other terms, like $(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc$. – wythagoras Aug 16 '15 at 06:34
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thanks. i came upon this while studying Graeff's root squaring method – Taylor Ted Aug 16 '15 at 06:36
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$$if\quad \quad { x }_{ 2 }-{ x }_{ 1 }=0\quad ,{ x }_{ 3 }-{ x }_{ 2 }=0,.......\quad etc\quad you\quad can\quad use\quad this\quad formula\quad \sum _{ i\quad =1 }^{ n }{ { i }^{ 3 } } =\quad { \left( \frac { n(n+1) }{ 2 } \right) }^{ 2 }\quad ,\quad n\quad ={ \quad x }_{ n }\\ Note*:\quad { \left( { x }_{ 1 }{ +x }_{ 2 }+{ x }_{ 3 }+....+{ x }_{ n } \right) }^{ 2 }\quad =\quad { { x }_{ 1 } }^{ 2 }+{ { x }_{ 2 } }^{ 2 }+{ { x }_{ 3 } }^{ 2 }+.....+{ { x }_{ n } }^{ 2 }+\\ \quad \quad \quad \quad \quad \quad \quad 2\left[ { x }_{ 1 }({ x }_{ 2 }+{ x }_{ 3 }+.......+{ x }_{ n })+{ x }_{ 2 }({ x }_{ 3 }+.......+{ x }_{ n })+...................+{ x }_{ n-1 }({ x }_{ n }) \right] $$
Science Addict
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Combining text with math is often not the best thing to do, but if you have to, you can use the \text command. For example: $\text{if }x_2-x_1=0, x_3-x_2=0, \dots\text{ etc., you can use this formula }\sum\limits_{i=1}^n i^3=$ which is typeset as
$\text{if }x_2-x_1=0, x_3-x_2=0, \dots\text{ etc., you can use this formula }\sum\limits_{i=1}^n i^3=$. – Martin Sleziak Aug 16 '15 at 08:59 -
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I should probably add also link to this post on meta: How can I text in between “dollar signs” on MSE. – Martin Sleziak Aug 16 '15 at 09:10
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You can think of the expression inside the square to be a simple $n$ dimensional polynomial. Squaring a polynomial is the same as convolving it's coefficients with itself.
mathreadler
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