Mathematical Induction PDF

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SH1712

Series and Mathematical Induction Sequence:  It is a list of things that are in order.  It follows a certain pattern or rule.  An infinite sequence 𝑎𝑛 is a function, whose domain is the set of positive integers. The function values, or terms, of the sequence are represented by 𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 , … , 𝑎𝑛 , …  𝑎1 is called the first term while 𝑎𝑛 is the 𝑛𝑡ℎ term.  Sequences, whose domain consists only of the first n positive integers, are called finite sequence. Series:  It is the sum of terms.  It is indicated with either the Latin capital letter “S” or the Greek letter that corresponds to S, known as Sigma “Σ”.  It is often designated by Sn where n represents the number of terms of the sequence being added.  Sn is often called an nth partial sum, since it can represent the sum of a certain “part” of a sequence. SUMS OF POWERS

𝑛

1. ∑ 1 = 𝑛 𝑘=1 𝑛

2. ∑ 𝑘 = 𝑘=1 𝑛

𝑛(𝑛 + 1) 2

3. ∑ 𝑘 2 = 𝑘=1 𝑛

4. ∑ 𝑘 3 = 𝑘=1

𝑛(𝑛 + 1)(2𝑛 + 1) 6 𝑛2 (𝑛 + 1)2 4

BINOMIAL THEOREM Key Property of Pascal Triangle: Every entry (other than 1) is the sum of the two (2) entries diagonally above it.

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Expanding (𝒂 + 𝒃)𝒏 using Pascal Triangle: Note: The top row, 1, is considered as the 0th row of the Pascal Triangle. Example: Expand the binomial (a + b) 5 using Pascal Triangle. Solution: Write each term in a way that the exponents of “a” is increasing while the exponents of “b” is decreasing. (a + b) 5 = a5 + a4b + a3b2 + a2b3 + ab4 + b5 Now, we will make use of the Pascal Triangle to know the coefficient of each term. Since n = 5, the 5th row of the Pascal Triangle will be used. 5th Row = 1

5

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(a + b) 5 = (1)a5 + (5)a4b + (10)a3b2 + (10)a2b3 + (5)ab4 + (1)b5 Therefore, the expansion of the binomial (a + b)5 is: (a + b) 5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 The Binomial Coefficient: Let 𝑛 and 𝑟 be nonnegative integers, 𝑟 ≤ 𝑛. The binomial coefficient is denoted by (𝑟𝑛)and is defined by: 𝑛 𝑛! ( )= 𝑟 𝑟! (𝑛 − 𝑟)!

Key Property of the Binomial Coefficients: For any nonnegative integers 𝑟 and 𝑘 with 𝑟 ≤ 𝑘, 𝑘 𝑘 𝑘+1 ( )+( ) =( ) 𝑟−1 𝑟 𝑟

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Relationship of Pascal Triangle and Binomial Coefficient:

The Binomial Theorem 𝑛 𝑛 𝑛 𝑛 𝑛 (𝑎 + 𝑏)𝑛 = ( ) 𝑎𝑛 + ( ) 𝑎𝑛−1 𝑏 + ( ) 𝑎𝑛−2 𝑏 2 + ⋯ + ( ) 𝑎𝑏 𝑛−1 + ( ) 𝑏 𝑛 𝑛 0𝑛 1 2 𝑛−1 𝑛

= ∑ ( ) 𝑎𝑛−𝑘 𝑏𝑘 𝑘 𝑘=0

References: Blitzer, R. (2014). Precalculus (5th ed.) Boston, Massachusetts: Pearson Education, Inc. Coburn, J. (2016). Pre-Calculus. 2 Penn Plaza, New York. McGraw Hill Education. Roberts, D. (n.d.). Practice with sigma notation and series. In RegentPrep. Retrieved from: http://www.regentsprep.org/regents/math/algtrig/atp1b/SigmaPractice.htm Stewart, J., Redlin, L., & Watson, S. (2014). Precalculus mathematics for calculus (7th ed). Boston, Massachusetts: Cengage Learning.

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