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Algebra
 Binomial Theorem
  General or (π‘Ÿ+1)th Term
  Euler's Proof
  Examples
  Examples
  Examples
  Examples
  Examples
  Examples
  Examples
  Examples
 Sources and References

Algebra

Binomial Theorem

(π‘Ž+𝑏)𝑛=π‘Žπ‘›+π‘›π‘Žπ‘›βˆ’1𝑏+𝑛(π‘›βˆ’1)2!π‘Žπ‘›βˆ’2𝑏2+𝑛(π‘›βˆ’1)(π‘›βˆ’2)3!π‘Žπ‘›βˆ’3𝑏3+β‹―

General or (π‘Ÿ+1)th Term

General or (π‘Ÿ+1)th term, 𝑛(π‘›βˆ’1)(π‘›βˆ’2)β‹―(𝑛+π‘Ÿβˆ’1)π‘Ÿ!π‘Žπ‘›βˆ’π‘Ÿπ‘π‘Ÿ or if 𝑛 be a positive integer. 𝑛!(π‘›βˆ’1)!π‘Ÿ!π‘Žπ‘›βˆ’π‘Ÿπ‘π‘Ÿ If 𝑏 be negative, the signs of the even terms will be changed.
If 𝑛 be negative the expansion reduces to (π‘Ž+𝑏)βˆ’π‘›=π‘Žβˆ’π‘›βˆ’π‘›π‘Žβˆ’π‘›βˆ’1𝑏+𝑛(𝑛+1)2!π‘Žβˆ’π‘›βˆ’2𝑏2βˆ’π‘›(𝑛+1)(𝑛+2)3!π‘Žβˆ’π‘›βˆ’3𝑏3+β‹― General term, (βˆ’1)π‘Ÿπ‘›(π‘›βˆ’1)(π‘›βˆ’2)β‹―(𝑛+π‘Ÿβˆ’1)π‘Ÿ!π‘Žβˆ’π‘›βˆ’π‘Ÿπ‘π‘Ÿ

Euler's Proof

Let the expansionof (1+π‘₯)𝑛 be called 𝑓(𝑛). Then it may be proved by Induction that the equation 𝑓(π‘š)×𝑓(𝑛)=𝑓(π‘š+𝑛) is true when π‘š and 𝑛 are integers, and therefore universally true; because the form of an algebraical product is not altered by changing the letters involved into fractional or negative quantities. Hence 𝑓(π‘š+𝑛+𝑝+β‹―)=𝑓(π‘š)×𝑓(𝑛)×𝑓(𝑝)Γ—β‹― Put π‘š=𝑛=𝑝=β‹― to π‘˜ terms, each equal β„Žπ‘˜, and the theorem is proved for a fractional index.
Again, put βˆ’π‘› for π‘š, thusm whatever 𝑛 may be, 𝑓(βˆ’π‘›)×𝑓(𝑛)=𝑓(0)=1, which proves the theorem for a negative index.
For the greatest term in the expansion of (π‘Ž+𝑏)𝑛, take π‘Ÿ = the integral part of (𝑛+1)π‘π‘Ž+𝑏 or (π‘›βˆ’1)π‘π‘Žβˆ’π‘, according as 𝑛 is positive or negative.
But if 𝑏 be greater than π‘Ž, and 𝑛 negative or fractional, the terms increase without limit.

Examples

Required the 40th term of 1βˆ’2π‘₯342 Here π‘Ÿ=39; π‘Ž=1; 𝑏=βˆ’2π‘₯3; 𝑛=42. the term will be 42!3!39!βˆ’2π‘₯339=βˆ’42β‹…41β‹…401β‹…2β‹…32π‘₯339

Examples

Required the 31st term of (π‘Žβˆ’π‘₯)βˆ’4 Here, π‘Ÿ=30; 𝑏=βˆ’π‘₯; 𝑛=βˆ’4. the term will be (βˆ’1)304β‹…5β‹…6β‹…β‹―β‹…30β‹…31β‹…32β‹…331β‹…2β‹…3β‹…β‹―β‹…30π‘Žβˆ’34(βˆ’π‘₯)30=31β‹…32β‹…331β‹…2β‹…3β‹…π‘₯30π‘Ž34

Examples

Required the greatest term in the expansion of 1(1+π‘₯)6 when π‘₯=1417. And 1(1+π‘₯)6=(1+π‘₯)βˆ’6. Here 𝑛=6, π‘Ž=1, 𝑏=π‘₯ in the formula (π‘›βˆ’1)π‘π‘Žβˆ’π‘=5Γ—14171βˆ’1417=2313 therefore π‘Ÿ=23, and the greatest term =(βˆ’1)235β‹…6β‹…7β‹…β‹―β‹…271β‹…2β‹…3β‹…β‹―β‹…23141723=βˆ’24β‹…25β‹…26β‹…271β‹…2β‹…3β‹…4141723

Examples

Find the first negative term in the expansion of (2π‘Ž+3𝑏)173. Take π‘Ÿ the first integer which makes π‘›βˆ’π‘Ÿ+1 negative; therefore π‘Ÿ>𝑛+1=173+1=623; therefore π‘Ÿ=7. The term will be 173β‹…143β‹…113β‹…83β‹…53β‹…23(βˆ’13)7!β‹…(2π‘Ž)13(3𝑏)7=βˆ’17β‹…14β‹…11β‹…8β‹…5β‹…2β‹…17!⋅𝑏7(2π‘Ž)13

Examples

Required the coefficient of π‘₯34 in the expansion of 2+3π‘₯2βˆ’3π‘₯2 (2+3π‘₯)2(2βˆ’3π‘₯)2=(2+3π‘₯)2(2βˆ’3π‘₯)βˆ’2=2+3π‘₯221βˆ’32π‘₯βˆ’2 =1+3π‘₯+94π‘₯21+23π‘₯2+2β‹…31β‹…23π‘₯22+β‹―+333π‘₯232+343π‘₯233+353π‘₯234+β‹― the tree terms last written being those which produce π‘₯34 after multiplying (1+3π‘₯+94π‘₯2); 94π‘₯2Γ—333π‘₯232+3π‘₯Γ—343π‘₯233+1Γ—353π‘₯234 giving for the coefficient of π‘₯34 in the result. 29743232+1023233+353234=3063232 The coefficient of π‘₯𝑛 will in like manner be 9𝑛32π‘›βˆ’2

Examples

To write the coefficient of π‘₯3π‘š+1 in the expansion of π‘₯2βˆ’1π‘₯22𝑛+1. The general term is (2𝑛+1)!(2𝑛+1βˆ’π‘Ÿ)!π‘Ÿ!π‘₯2(2π‘›βˆ’π‘Ÿ+1)β‹…1π‘₯2π‘Ÿ=(2𝑛+1)!(2𝑛+1βˆ’π‘Ÿ)!π‘Ÿ!π‘₯4π‘›βˆ’4π‘Ÿ+2) Equate 4π‘›βˆ’4π‘Ÿ+2 to 3π‘š+1, thus π‘Ÿ=4π‘›βˆ’3π‘š+14. Substitute this value of π‘Ÿ in the general term; the required coefficient becomes (2𝑛+1)![ΒΌ(4𝑛+3π‘š+1)]![ΒΌ(4π‘›βˆ’3π‘š+1)]! The value of π‘Ÿ shows that there is no term in π‘₯3π‘š+1 unless 4π‘›βˆ’3π‘š+14 is an integer.

Examples

An approximate value of (1+π‘₯)𝑛, when π‘₯ is small, is 1+𝑛π‘₯, by neglecting π‘₯2 and higher powers of π‘₯.

Examples

An approximation to 3999 by obtaining from the first two or three terms of the expansion of (1000βˆ’1)12=10βˆ’23β‹…1000βˆ’13=10βˆ’1300=900299300 nearly

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive

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ID: 210600010 Last Updated: 6/10/2021 Revision: 0 Ref:

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References

  1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
  2. Wheatstone, C., 1854, On the Formation of Powers from Arithmetical Progressions
  3. Stroud, K.A., 2001, Engineering Mathematics
  4. Coolidge, J.L., 1949, The Story of The Binomial Theorem
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