# The Mathematical Ninja and the Unknown Powers

The Mathematical Ninja peered at the problem sheet:

Given that $(1+ax)^n = 1 - 12x + 63x^2 + \dots$, find the values of a and n

Barked: “$n=-8$ and $a=\frac{3}{2}$.”

The student sighed. “I get no marks if I just write down the answer.”

Snarled: “You get no scars if you don’t talk back.”

“Are you going to teach me your secrets, oh all-knowing sensei?” The Mathematical Ninja, the student had learned, had one weakness: the inability to detect sarcasm.

“$n$ is simply $\frac{2\times 63}{(-12)^2 - 126}$, which is 8.”

“No, you’ll need to run through that a bit more slowly.”

Under their breath: “Only one thing getting run through around here, sunshine, and the only slow thing will be your demise.” Aloud: “Fine, fine, fine. I shall break it down into simpler slices.” Slices. Now there was a thought.

“You’re too kind, sensei.”

“We know that $an = -12$ and $\frac{1}{2}a^2n(n-1) = 63$, because even you understand the binomial expansion.”

The student pretended this was the case.

“If I square the first and double the second, I get $a^2 n^2 = 144$ and $a^2n(n-1) = 126$.”

“Which you can divide to get rid of the $a$?”

“Indeed. $\frac{n(n-1)}{n^2}$ - or rather, $\frac{n-1}{n} = \frac{126}{144}$.”

“And you could simplify that to $\frac{7}{8}$ and read off the answer?”

“In this case, yes, but generally you might not get a nice fraction. In the more general case, if you had $1 - Ax + Bx^2 + \dots$, you would end up with $\frac{n-1}{n} = \frac{2B}{A^2}$. That’s equivalent to $1-\frac{1}{n} = \frac{2B}{A^2}$, or $\frac{1}{n} = \frac{A^2 - 2B}{A^2}$.”

“And you can flip that to get your answer!”

The Mathematical Ninja restrained themself from flipping a table. “$n$ is indeed $\frac{A^2}{A^2 - 2B}$, yes. And here, that’s $\frac{144}{144-126}$; and everyone knows that $144$ is eight eighteens.”

“Everyone, sensei. Absolutely everyone.”

* Edited 2019-08-26 after the Ninja pointed out that I'd mis-transcribed their original answer. I am typing this with nine fingers. It won't happen again. ## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

#### Share

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Sign up for the Sum Comfort newsletter and get a free e-book of mathematical quotations.

No spam ever, obviously.

##### Where do you teach?

I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.
{"ft_lang_label":"__label__en","ft_lang_prob":0.8871251,"math_prob":0.98608756,"size":2028,"snap":"2019-35-2019-39","text_gpt3_token_len":635,"char_repetition_ratio":0.10326087,"word_repetition_ratio":0.0,"special_character_ratio":0.3500986,"punctuation_ratio":0.1403118,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9975633,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-17T10:37:29Z\",\"WARC-Record-ID\":\"<urn:uuid:20f46dc5-362f-478f-a34f-b6cc17802ff8>\",\"Content-Length\":\"58332\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5060eae0-e198-4612-bcd5-9460a3b0ee0b>\",\"WARC-Concurrent-To\":\"<urn:uuid:9141e941-b57c-4896-83f1-0a0aad6970d4>\",\"WARC-IP-Address\":\"185.20.49.4\",\"WARC-Target-URI\":\"http://www.flyingcoloursmaths.co.uk/the-mathematical-ninja-and-the-unknown-powers/\",\"WARC-Payload-Digest\":\"sha1:K5YMBAS5DMMPHIJKCATAZI34QPE26NG3\",\"WARC-Block-Digest\":\"sha1:CP3FJNUEDYQKZWFGJSKTJRBZYF7ODHGN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573070.36_warc_CC-MAIN-20190917101137-20190917123137-00506.warc.gz\"}"}