Show That C1 + C2 + C3 + .... Cn = 2n − 1 - Mathematics and Statistics

Show That C₁ + C₂ + C₃ + .... C_n = 2ⁿ − 1

Sum

Since C₀ + C₁ + C₂ + C₃ + ……+ C_n = 2ⁿ

But, C₀ = 1

∴ 1 + C₁ + C₂ + C₃ + …… + C_n = 2ⁿ

∴ C₁ + C₂ + C₃ + …… + C_n = 2ⁿ – 1

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Binomial Theorem for Negative Index Or Fraction

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Chapter 4: Methods of Induction and Binomial Theorem - Exercise 4.5 [Page 84]

Chapter 4 Methods of Induction and Binomial Theorem
Exercise 4.5 | Q 6 | Page 84

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 + x)⁻⁴

State, by writing first four terms, the expansion of the following, where |x| < 1

`(1 - x)^(1/3)`

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 – x²)^–3

State, by writing first four terms, the expansion of the following, where |x| < 1

`(1 + x)^(-1/5)`

State, by writing first four terms, the expansion of the following, where |x| < 1

(1 + x²)^–1

State, by writing first four terms, the expansion of the following, where |b| < |a|

(a − b)⁻³

State, by writing first four terms, the expansion of the following, where |b| < |a|

(a + b)⁻⁴

State, by writing first four terms, the expansion of the following, where |b| < |a|

`("a" + "b")^(1/4)`

State, by writing first four terms, the expansion of the following, where |b| < |a|

`("a" - "b")^(-1/4)`

State, by writing first four terms, the expansion of the following, where |b| < |a|

`("a" + "b")^(-1/3)`