My Math Forum Binomial Theorem! stuck again! :S

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 January 31st, 2012, 08:04 AM #1 Member   Joined: Dec 2011 Posts: 75 Thanks: 0 Binomial Theorem! stuck again! :S I'm stuck again! :s please can anyone help!, I start off fine, thinking I've got a result then it asks to express as a fraction, by my answer doesn't go into one. so I'm doing something wrong. Use the Binomial Theorem to determine the coefficient of x^18 in the expansion of (1/14x^2 - 7)^16 Express the coefficient as a fraction in its lowest terms.
January 31st, 2012, 08:46 AM   #2
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Re: Binomial Theorem! stuck again! :S

Hello, fe phi fo!

Quote:
 $\text{Use the Binomial Theorem to determine the coefficient of }x^{18}$ [color=beige]. . [/color]$\text{ in the expansion of: }\:\left(\frac{1}{14}x^2\,-\,7\right)^{16}$

$\text{The expansion of }(a\,-\,b)^{16}\text{ begins:}$

[color=beige]. . [/color]$a^{16}\,-\,16a^{15}b\,+\,120a^{14}b^2\,-\,560a^{13}b^3\,+\,1820a^{12}b^4\,-\,4368a^{11}b^5\,+\,8008a^{10}b^6 \,-\,11,440a^9b^7\,+\,\text{ . . .}$

$\text{W\!e have: }\:a= \frac{x^2}{14},\;b = 7$

$\text{The term with }x^{18}\text{ would be: }-11,440a^9b^7$

$\text{W\!e have: }\:-11,440\,\!\left(\frac{x^2}{14}\right)^9\big(7\big) ^7 \;=\;\frac{(-11,440)(7^7)}{14^9}\,x^{18}$

$\text{The coefficient is: }\:-\frac{(11,440)(7^7)}{(14^2)(14^7)} \;=\; -\left(\frac{11,440}{14^2}\right)\left(\frac{7^7}{1 4^7}\right) \;=\;-\left(\frac{11,440}{14^2}\right)\left(\frac{7}{14} \right)^7 \;=\;-\left(\frac{11,440}{14^2}\right)\left(\frac{1}{2}\ right)^7$

[color=beige]. . . . . . . . . . . . . [/color]$=\;-\frac{11,440}{(196)(128)} \;=\;-\frac{11,440}{25,088} \;=\;-\frac{715}{1568}$

 January 31st, 2012, 09:42 AM #3 Member   Joined: Dec 2011 Posts: 75 Thanks: 0 Re: Binomial Theorem! stuck again! :S Hello again soroban! Thanks again for your quick response! and the explanation which was great too, really helpful, (I'm just glad that I'm on the right track! one day the penny will drop! ) I just couldn't figure out how you were suppose to turn the 18th term which you got into a fraction, which threw me! (-11440)(7^7)/14^9 x^18 - very helpful! there's just one thing i don't see how you got there, as the result i got was 11440a^9b^7 but positive, because when I look at the binomial theorem, each term is added to the previous. how and why did your result become negative and have the sequence + then - then + etc ..
 January 31st, 2012, 10:14 AM #4 Member   Joined: Dec 2011 Posts: 75 Thanks: 0 Re: Binomial Theorem! stuck again! :S Hey! Sorry i figured out why you obtained a negative -11440, i was using the calculator ha ha which was obviously giving me a positive factorial, it just shows its helps to right out the sequence fully! thanks again for your help! till next time

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