Binomial

Here is the onsite calculator output.....

2^7+7*(2^6)*(.3)+21(2^5)*(.3)^2+35(2^4)*(.3)^3+35(2^3)*(.3^4)+21*(2^2)*(.3)^5+7*(2)*(.3^6)+(.3^7)

=  340.4825447     

I didn't seem to have any problems with it...... (???)

The binomial expansion of  

$$\\(a+b)^7= \sum \limits_{n=0}^7} \;^7C_n\;b^n\;a^{(7-n)}\\\\\\
so\\\\
2.3^7 \;= \;(2+0.3)^7= \sum \limits_{n=0}^7} \;^7C_n\;(0.3)^n\;2^{(7-n)}\\\\$$

Does that help?

They just rounded the last answer (in red) from the blue one before it....."+smaller terms" means they just didn't list all of the evaluated terms..... if you evaluate all the first line of stuff......I'm thinking it should = that "blue" answer

Yes that sounds good Chris :))

I keep getting 340.47212 on my calculator though when I added all the numbers up. Wouldn't it be 340.472? Am I missing a number or step? 

(2.3)^7=(2+0.3)^7

$${{\mathtt{2}}}^{{\mathtt{7}}} = {\mathtt{128}}$$

$${\mathtt{7}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{6}}}{\mathtt{\,\times\,}}{\mathtt{0.3}} = {\frac{{\mathtt{672}}}{{\mathtt{5}}}} = {\mathtt{134.4}}$$

$${\left({\frac{{\mathtt{7}}{!}}{{\mathtt{2}}{!}{\mathtt{\,\times\,}}({\mathtt{7}}{\mathtt{\,-\,}}{\mathtt{2}}){!}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{{\mathtt{0.3}}}^{{\mathtt{2}}} = {\frac{{\mathtt{1\,512}}}{{\mathtt{25}}}} = {\mathtt{60.48}}$$

$${\left({\frac{{\mathtt{7}}{!}}{{\mathtt{3}}{!}{\mathtt{\,\times\,}}({\mathtt{7}}{\mathtt{\,-\,}}{\mathtt{3}}){!}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{{\mathtt{0.3}}}^{{\mathtt{3}}} = {\frac{{\mathtt{378}}}{{\mathtt{25}}}} = {\mathtt{15.12}}$$

$${\left({\frac{{\mathtt{7}}{!}}{{\mathtt{4}}{!}{\mathtt{\,\times\,}}({\mathtt{7}}{\mathtt{\,-\,}}{\mathtt{4}}){!}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{3}}}{\mathtt{\,\times\,}}{{\mathtt{0.3}}}^{{\mathtt{4}}} = {\frac{{\mathtt{567}}}{{\mathtt{250}}}} = {\mathtt{2.268}}$$

$${\left({\frac{{\mathtt{7}}{!}}{{\mathtt{5}}{!}{\mathtt{\,\times\,}}({\mathtt{7}}{\mathtt{\,-\,}}{\mathtt{5}}){!}}}\right)}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{{\mathtt{0.3}}}^{{\mathtt{5}}} = {\mathtt{0.204\: \!12}}$$

$${\mathtt{7}}{\mathtt{\,\times\,}}{{\mathtt{2}}}^{{\mathtt{1}}}{\mathtt{\,\times\,}}{{\mathtt{0.3}}}^{{\mathtt{6}}} = {\mathtt{0.010\: \!206}}$$

$${{\mathtt{0.3}}}^{{\mathtt{7}}} = {\mathtt{0.000\: \!218\: \!7}}$$

total = $${\mathtt{128}}{\mathtt{\,\small\textbf+\,}}{\mathtt{134.4}}{\mathtt{\,\small\textbf+\,}}{\mathtt{60.48}}{\mathtt{\,\small\textbf+\,}}{\mathtt{15.12}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2.268}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.204\: \!12}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.010\: \!206}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.000\: \!218\: \!7}} = {\mathtt{340.482\: \!544\: \!7}}$$

Ohh I know what I did wrong. I forgot to move the decimal places in 7x2x0.3^6

Thanks ya'll!

Yeah....I  kinda' thought it would either be a "decimal issue" or that you possibly didn't include all the terms in your summation......glad we could help ya'.......