Show that
C₀ + C₂ + C₄ + C₆ + C₈ = C₁ + C₃ + C₅ + C₇ = 128
50 Points
Answers
Answer:
Consider the expansion of (1 + x)⁸ = C₀ + C₁x + C₂x² + C₃x³ + C₄x⁴ + C₅x⁵
+ C₆x⁶ + C₇x⁷ + C₈x⁸-------------------(1)
Substituting x = -1 in the above expression , we get
(1 - 1)⁸ = C₀ - C₁ + C₂ - C₃ + C₄ - C₅ + C₆ - C₇ + C₈
⇒C₀ - C₁ + C₂ - C₃ + C₄ - C₅ + C₆ - C₇ + C₈ = 0
⇒C₀ + C₂ + C₄ + C₆ + C₈ = C₁ + C₃ + C₅ + C₇ -------(2)
Substituting x = 1 in (1), we get
(1 + 1)⁸ = C₀ + C₁ + C₂ + C₃ + C₄ + C₅ + C₆ + C₇ + C₈
⇒C₀ + C₁ + C₂ + C₃ + C₄ + C₅ + C₆ + C₇ + C₈ = 256
⇒(C₀ + C₂ + C₄ + C₆ + C₈) + (C₁ + C₃ + C₅ + C₇) = 256
Using (2), we get
C₀ + C₂ + C₄ + C₆ + C₈ = (C₁ + C₃ + C₅ + C₇) = 128
Step-by-step explanation:
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Step-by-step explanation:
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.Numbers starting from 1, 2, 3, 4, … and so on are known as natural numbers. A group of digits together forms a number where the digits can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
1. There are two methods of representing a number:
a) Indian System of Numeration
b) International System of Numeration