Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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### Mathematical Identity **5. If \( n \in \mathbb{N} \), then \( 2^1 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 2 \).** This equation represents a mathematical identity that expresses the sum of powers of two. The left side of the equation is the summation of powers of two starting from \( 2^1 \) up to \( 2^n \). The right side of the equation simplifies this summation to a single expression: \( 2^{n+1} - 2 \). This identity can be proved using mathematical induction or by recognizing it as a specific case of the geometric series sum formula.
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Transcribed Image Text:### Mathematical Identity **5. If \( n \in \mathbb{N} \), then \( 2^1 + 2^2 + 2^3 + \cdots + 2^n = 2^{n+1} - 2 \).** This equation represents a mathematical identity that expresses the sum of powers of two. The left side of the equation is the summation of powers of two starting from \( 2^1 \) up to \( 2^n \). The right side of the equation simplifies this summation to a single expression: \( 2^{n+1} - 2 \). This identity can be proved using mathematical induction or by recognizing it as a specific case of the geometric series sum formula.
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