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I have a question that I can solve quite nicely with the binomial theorem; however, I am required to use pascals triangle for a lower level course:

In the expansion of $(1+x)^n$ the first three terms are $1$, $-18$, and $144$. Determine the values of $x$ and $n$.

Answer: $x=-2$, $n=9$

But I don't know how to solve this using pascals triangle!

Thanks

Gary
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Dodd-learning
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    Hint: Write the expansion of $(1+x)^n$ and see if you can make any associations. – Sean Roberson Apr 19 '23 at 02:58
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    Pascal's triangle only tells you $\binom n k$, mind you. These are the coefficients of $x^k$ in $(1+x)^n$, but they are not the entire term. Just bear that in mind. – PrincessEev Apr 19 '23 at 03:06
  • @SeanRoberson Thanks for the suggestion. How would I go about writing the expansion of (1+x)^n? I've only ever done expansion using pascals triangle when n is defined. Not sure what row to look at here! Thanks – Dodd-learning Apr 19 '23 at 03:56
  • Each row tells your the coefficients of an expansion. You'll need to use that as a guide. – Sean Roberson Apr 19 '23 at 03:58
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    Hint: The coefficients of the first three terms are $~\displaystyle \binom{n}{0}, ~\binom{n}{1}, ~\binom{n}{2}.$ First, solve for $~n,~$ and then solve for $~x.$ – user2661923 Apr 19 '23 at 04:06
  • What's the difference between Pascal's triangle and the binomial theorem, except for the format? – Raskolnikov Apr 19 '23 at 07:58
  • Thanks everyone. I got it by justifying pattern recognition (aka the binomial theorm). – Dodd-learning Apr 20 '23 at 17:31

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