# inductive proof of binomial theorem

We prove the theorem for a ring. We do not assume a unit for the ring. We do not need commutativity of the ring, but only that $a$ and $b$ commute.

When $n=1$, the result is clear.

For the inductive step, assume it holds for $m$. Then for $n=m+1$,

 $\displaystyle(a+b)^{m+1}$ $\displaystyle=$ $\displaystyle(a+b)(a+b)^{m}$ $\displaystyle=$ $\displaystyle(a+b)(a^{m}+b^{m}+\sum_{k=1}^{m-1}\binom{m}{k}a^{m-k}b^{k})\text{% by the inductive hypothesis}$ $\displaystyle=$ $\displaystyle a^{m+1}+b^{m+1}+ab^{m}+ba^{m}+\sum_{k=1}^{m-1}\binom{m}{k}a^{m-k% +1}b^{k}+\sum_{k=1}^{m-1}\binom{m}{k}a^{m-k}b^{k+1}$ $\displaystyle=$ $\displaystyle a^{m+1}+b^{m+1}+\sum_{k=1}^{m}\binom{m}{k}a^{m-k+1}b^{k}+\sum_{k% =0}^{m-1}\binom{m}{k}a^{m-k}b^{k+1}\text{ by combining terms}$ $\displaystyle=$ $\displaystyle a^{m+1}+b^{m+1}+\sum_{k=1}^{m}\binom{m}{k}a^{m-k+1}b^{k}+\sum_{j% =1}^{m}\binom{m}{j-1}a^{m+1-j}b^{j}\text{ let j=k+1 in second sum}$ $\displaystyle=$ $\displaystyle a^{m+1}+b^{m+1}+\sum_{k=1}^{m}\left[\binom{m}{k}+\binom{m}{k-1}% \right]a^{m+1-k}b^{k}\text{ by combining the sums}$ $\displaystyle=$ $\displaystyle a^{m+1}+b^{m+1}+\sum_{k=1}^{m}\binom{m+1}{k}a^{m+1-k}b^{k}\text{% from Pascal's rule}$

as desired.

Title inductive proof of binomial theorem InductiveProofOfBinomialTheorem 2013-03-22 11:48:06 2013-03-22 11:48:06 Mathprof (13753) Mathprof (13753) 21 Mathprof (13753) Proof msc 05A10
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 20, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9763475060462952, "perplexity": 500.4976340945635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583514879.30/warc/CC-MAIN-20181022071304-20181022092804-00554.warc.gz"}