C2 sequences and series binomial expansion answers

Example 5. where k is a constant. (i) (ii) (iii) How did you do? View Answer. 1. 1st class MSci Astrophysics. Find the first 3 terms, in ascending powers of x, of the binomial expansion of. f(x) =3+5𝑥𝑥 (1+3𝑥𝑥)(1+𝑥𝑥)2. ( a + b) 5. Experienced Mathematics Tutor for GCSE, A levels and IB. May 17, 2017 · Divergence simply means "not convergence". Find the first 3 terms, in ascending powers of x, of the binomial expansion of binomial theo - Free download as PDF File (. 6352 (NP: If it only mentions to use y terms in the question, you only need to add together y terms in the answer) C2 Sequences & Series: Binomial Expansion PhysicsAndMathsTutor. Graduate. (i) Given that . 1 Binomial expansion 1. x) + 6(1 2. Sequences and Series Key Skills Section (for selecting more than one) Jan 24, 2012 · The Sequence and Series chapter in c2, is quite big, so I will divide it into 3 / 4 posts. Find an expression for b in terms of a. Find the first 4 terms of the binomial expansion, in ascending powers of x, of :1+𝑥 4 ;8 giving each term in its simplest form. 001) + 115200(0. This expansion is only valid when. [2] 10. x . (2) (Total 6 marks) 4. When we take the sum of the terms in a sequence, we get a series. 97468099. 9. k. (4) Given that the coefficient of x2 is 6 times the coefficient of x, (b) find the value of k. C2-Sequences-Series-C-Simple-Binomial-Expansion-Answers. Show Step-by-step Solutions Binomial Expansion 1a. 005⁹ --> ∴ x = 0. (a) Find the first 4 terms, in ascending powers of x, of the binomial expansion of (1 + ax)7, where a is a constant. Binomial Expansion Series, Sequences, and Binomial Expansion Test Calendar Subject to Change! HW 1: HW 7: Answer all questions in #1 – 6: 1. P(r / n) = ∑ni = 0Cniqn − ipiδir = Cnrqn − rpr. May 1, 2024 · Then, Binomial distribution is a Taylor expansion of a binomial (q + p)n where n (# of trials) is the order of interest, and r (# of successes) is the parameter defining the dominating term through. < 1 35 terms of the series. a n. Binomial Expansion 1a. P2 | C12 | C2. 23 Use the Binomial Theorem to Expand a Binomial. in ascending powers of . Define geometric sequence as a sequence in which each term after the first is found by multiplying the preceding term by a constant ratio. Dec 11, 2010 · C2 Sequences Series: Binomial Expansion PhysicsAndMathsTutor. The first three terms in the expansion of (1 + ax)b, in ascending powers of x, for |𝑎𝑎| < 1𝑥𝑥, are 1 – 6x+ 24x2. a Find the first 4 terms, in ascending powers of x, of the binomial expansion&#8230; In the binomial expansion of (1 + x)40, the coefficients of x4 and x5 are p and q respectively. 001 2) Substitute in the x value into the Expansion. Download these Free Sequences and Series MCQ Quiz Pdf and prepare for your upcoming exams Like Banking, SSC, Railway, UPSC, State PSC. As a farmer bales a Transcript. Some of the worksheets displayed are Binomial expansion work, The binomial expansion, Binomial expansion question work, Sequences and series part 1b binomial expansion, The binomial theorem, Binomial expansions exam questions, Work the binomial theorem, C2 the binomial theorem work Maths/Physics Examiner Who Has Helped 6 GCSE/IB & 8 A Level Students Acheive A*'s In Last Year Alone. The Binomial theorem tells us how to expand expressions of the form (a+b)ⁿ, for example, (x+y)⁷. 034 642 080 = 527 205. 1b 2 C2 2017 1 4. 528 + 15. 13 a Expand (3 – 3 x)12 as a binomial series in ascending powers of x up to and including the term in x3, giving each coefficient as an integer. 97 10. 2. 1 −. . Answer: 2, −4, 8, −16, 32. All C4 Revsion Notes. The binomial coefficients are the integers calculated using the formula: (n k) = n! k!(n − k)!. Sep 19, 2022 · Help Center Detailed answers to any questions you might have Using the binomial expansion for $ sequences-and-series; Feb 14, 2022 · The real beauty of the Binomial Theorem is that it gives a formula for any particular term of the expansion without having to compute the whole sum. So the term containing x6 is. 14 a Expand (1 – x)5 as a binomial series in ascending powers of x. 1 Into a candidate's binomial expansion. Infinite Geometric Series , where we add all of the terms in the geometric progression. a. are both positive, show that . Consider a geometric sequence with kth term a k = ark such that: a 1 = 1; X1 k=0 ark = 9 2: (a)Either a = 3 and r = 1 3, or a = 3 2 and r = 2 3. 528 Chapter 8 Sequences , Series , and Probability. (a) Find the first 4 terms, in ascending powers of x, of the binomial… In the binomial expansion of (k + ax)4 the coefficient of x2 is 24. x. 4. where P(r / n) is the probability to observe the event r / n of r successes out of n trials. 2 = (1 − 2 )1 2. (9−x)4 ( 9 − x) 4 Solution. [4] Showing top 8 worksheets in the category - Binominal Expansion. Let’s take a quick look at an example. [5] 11. 15. Questions and answers with explanations on binomial theorems Sequences and series - Binomial series PhysicsAndMathsTutor. The binomial expansion Exercise A, Question 6 © Pearson Education Ltd 2008 Question: The coefficient of x2 in the expansion of ( 2 + ax ) 3 is 54. and for the cube of a binomial, we obtain after expanding. E: Sequences, Series, and the Binomial Theorem (Exercises) is shared under a CC BY-NC-SA 3. Prove that S = Xn k=1 a k = 1 2 n(2a+ (n 1)d): [8] Hint: this is a proof that you may have seen in class. 588 936 − 0. 792 Binomial Expansion of (ax±b) n, Where n is a Positive Integer. x , } expansion With candidate' s followed through ( ** x) Award SC Ml if you see Either 2 {1. To calculate the 50th partial sum of this sequence we need the 1st and the 50th terms: a1 = 4 a50 = 5 − 1 = 249. x, of the binomial expansion 5. Experienced, full-time, online tutor. 99812 ≈ 531 441 − 4251. x)12 as a binomial series in ascending powers of x up to and including the term in x3, giving each coefficient as an integer. Find the value of (2) 4a. 0000065104166. $\endgroup$ – Oct 11, 2016 · It is not currently accepting answers. 1 anything that cancels to 2 Simplified —Xx2 — x Attempt to substitute 0. [2] (b) x3 17. (2) b. Show Solution. In the following exercises, find the coefficient of the indicated term in the expansion of the binomial. in the expansion is 128, find the values of . (2) Given that the third term of this series is 540x 2 , (b) show that k = 6, (2) (c) find the coefficient of x 3. This section looks at Binomial Theorem and Pascals Triangle. It should also be obvious to you that (a + b)¹ = a + b . 3√8−2x 8 − 2 x 3 Solution. (a) Find the value of p and the value of q. (a) Find the first 4 terms, in ascending powers of x, of the binomial… AS and A level Mathematics Practice Paper – Binomial expansion – Mark scheme 5 Source paper Question number New spec references Question description New AOs 1 C2 2012 1 4. + 9x + px2 + qx3, 12x < 1. The exponents on a decrease by one on each term going left to right. 𝑦𝑦 We can either use the binomial formula or Pascal’s triangle to expand expressions of the form (𝑎𝑎+ 𝑏𝑏)𝑛𝑛. C2 SEQUENCES AND SERIESAnswers - Worksheet C. Answer. 4. 025)8, giving your answer to 4 The third term of a geometric sequence is 324 and the sixth term is 96 (a) Show that the common ratio of the sequence is . The New 2017 A level page. Formula Book. 1 Infinite Sequences 401 8. 0 license and was authored, remixed, and/or curated by Carl Stitz & Jeff Zeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. e= 1 + 4(1 2. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. (b) Use your expansion to estimate the value of (1. r. co. In the row below, row 2, we write two 1’s. Oct 6, 2021 · This page titled 9. 1 3 marks. b n. in this expansion is 525, (b) find the possible values of C2 - Sequences and Series OCR, AQA, Edexcel 1. All C1 Revsion Notes. 5E. Find the common difference and the first term. The Binomial Expansion, is a theorem which allows us to expand (a + b)^n, where n is an integer. Find the value of p and the value of q. You should know that (a + b)² = a² + 2ab + b² and you should be able to work out that (a + b)³ = a³ + 3a²b + 3b²a + b³ . This topic is included in all papers for AS-level and A-level OCR (MEI) Maths. 1 Key Facts: Informal Binomial Expansion A binomial is a polynomial that is the sum of two terms (e. 03 (2dp) an = a1 + (n − 1) d = 4 + (n − 1) ⋅ 5 = 4 + 5n − 5 = 5n − 1. (4) (b) Use this expansion with your values of p and q together with an appropriate value of x to obtain an estimate of (1. the Binomial Theorem 8. Thus, if we denote the terms of a binomial by a and b, the square of a binomial gives after expanding it. txt) or read online for free. London Science College © 2024 Geometric Progressions, where we multiply by a fixed number to get each new term of the progression. For example. Use your expansion to estimate the value of (1. Questions are taken from the pre 2010 exam papers. b) Use the first three terms in the binomial expansion of ( )2 3− x 10, with a suitable value for x, to find an approximation for 1. 1 Binomial Expansion for the OCR A Level Maths: Pure syllabus D1-1 9 Binomial Expansion: EXTENSION Extending Binomial Expansion D1- 20 Binomial Expansion: Writing (a + bx)^n in the form p(1 + qx)^n D1- 21 Binomial Expansion: Find the first four terms of (1 + x)^(-1) All A level questions arranged by topic. p5q4 term of (3p + q)9. 2 + x k , where k is a constant. 1 Binomial Expansion for the OCR A Level Maths: Pure syllabus, written by the Maths experts at Save My Exams. 2: Arithmetic Sequences and Series; 9. Notice, that in each case the exponent on the \(b\) is one less than the number of the term. x7 term of (x − 3)9. The first term is an. 𝑎𝑎𝑎𝑎+ 𝑏𝑏). C2 Sequences and Series. 99812, giving your answer to 2 decimal places. Model Answers. To generate Pascal’s Triangle, we start by writing a 1. 10 Arithmetic Series: Finding a and d. and the last term is bn. Key Skills. The binomial expansion of . (4) (b) Use this expansion with your values of p and q together with an appropriate value of x = 4. Expand the following expressions. 1. The sum of the exponents on any term is n. Jan 3, 2023 · A sequence is simply a list of numbers in a particular order. g. Jan 2009 qu. Let’s look for a pattern in the Binomial Theorem. And so we get the answer: X1 k=1 4 1 6 k = 24 5 4 = 4 5 [4] 10. x, of the binomial expansion Free Binomial Expansion Calculator - Expand binomials using the binomial expansion method step-by-step Apr 6, 2018 · C2 Sequences & Series: Binomial Expansion PhysicsAndMathsTutor. giving each term in its simplest form. pdf), Text File (. 006 2. a4b2 term of (2a + b)6. [3] (ii) Given also that the coefficient of . k . SEND. ξ1 − 2 2, Example 1: Find the expansion of up to and including the term in and state of values for for which the expansion is valid. 3 (i) Find and simplify the first four terms in the binomial expansion of (1 + x) 10. aectutors. 0003125 1. 1 Binomial Expansion for the Edexcel A Level Maths: Pure a) Determine, in ascending powers of x, the first three terms in the binomial expansion of ( )2 3− x 10. (a) Find the first 4 terms of the binomial expansion, in ascending powers of x, of (1 + x/4)8 giving each term in its simplest form. Consider an arithmetic sequence with kth term given by a k = a+ (k 1)d. (1+3x)−6 ( 1 + 3 x) − 6 Solution. b the value of the coefficient of x3 in the expansion. For instance, 2,4,6,8 are the first four terms in the sequence of even positive integers. We denote the terms in a sequence by Video answers for all textbook questions of chapter 14, Binomial Expansions, Sequences, and Series, Beginning and Intermediate Algebra by Numerade Get 5 free video unlocks on our app with code GOMOBILE C2 SEQUENCES AND SERIES Answers - Worksheet C 1 4a = 1 + 4x + 6x 2 + 4x3 + x b = 1 − 5x + 10x − 10x3 + 5x4 − x5 c = 1 + 3(4x) + 3(4x)2 + (4x)3 d = 1 + 3(−2y) + 3(−2y)2 + (−2y)3 = 1 + 12x + 48x2 + 64x3 = 1 − 6y + 12y2 − 8y3 e = 1 + 4(1 2 x) + 6(1 2 x)2 + 4(1 2 x)3 + (1 2 x)4 f = 1 + 3(1 3 y) + 3(1 3 y)2 + (1 3 y)3 = 1 + 2x + 3 2 C2 SEQUENCES AND SERIES Answers - Worksheet A 1 a r = 3 b r = 1 4 c r = −2 u8 = 3 × 3 7 = 6561 u 8 = 1024 (a) Find the first four terms, in ascending powers of x, in the binomial expansion of 5. x)2+ 4(1 2. For problems 3 and 4 write down the first four terms in the binomial series for the given function. x5 term of (x − 4)6. The larger the power is, the harder it is to expand expressions like this directly. 1 Binomial Expansion for the AQA A Level Maths: Pure syllabus C2 SEQUENCES AND SERIES Answers - Worksheet D page 4 Solomon Press 13 a12 + 12(3 = 311)(− 3 x) + 12 11 2 × (310)(− 3 x)2 + 12 11 10 32 ×× × (39)(− 3 x)3 + … = 531 441 − 708 588x + 433 026x2 − 160 380x3 + … b let 3 x = 0. 5 Find the coefficient of x6 in the expansion of (3 + 2x) 10. 4 Infinite Figure 2. In the [latex]n\text {th} [/latex] row, flank the ends of the row with 1’s. 1 Binomial Expansion for the AQA A Level Maths: Pure syllabus, written by the Maths experts at Save My Exams. Find the series expansion of f(x) in ascending powers of xup to and including the term in x3and state the set of value of x for which it is valid. 1 9746810 2 0. In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000. Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost. Rewriting so the power is visible. We call these numbers the terms of the sequence. 3: Geometric Sequences and Series A geometric sequence, or geometric progression, is a sequence of numbers where each successive number is the product of the previous number and some constant r . Solution: Let us take a = 3 and b = 2x in the binomial expansion of (a + b) 10. com Edexcel Internal Review 1 . 1b 17. Now, the Binomial Theorem required that n n be a positive integer. You can find Edexcel International A-level P2 (WMA12), C12 (WMA01), and Edexcel A-level old spec C2 (6664), past papers, mark schemes and model answers below: expand. 17. In the 3 rd row, flank the ends of the rows with 1’s, and add [latex]1+1 [/latex] to find the middle number, 2. A sequence (or series) is divergent if and only if it is not convergent. (2) (Total 9 marks) (b) The first four terms of the binomial expans ion of in ascending powers of x are 1 + ax + bx 2 + cx 3. June 2010 qu. Sequences and series - Binomial series PhysicsAndMathsTutor. Solution: ( 2 +x a ) 3 has coefficients 1 3 1 The circled number is the coefficient of the term 21 (x a ) 2. Dec 27, 2014 · C2 Sequences & Series: Binomial Expansion 1. 14a= 1 + 4x+ 6x2+ 4x3+ xb= 1 − 5x+ 10x− 10x3+ 5x4−x5. The Binomial Theorem, where we learn how to expand expressions like. Therefore, the general term is expressed in terms of the previous two as follows: F n = F n − 2 + F n − 1. Follows correct answer with 27 90x+120x2 can iswhere (sp marks for correct answer Misreads ascending and gives —32x5 + 240x4 — 720x3 is marked as BIBOMIAO special case and must be completely correct (If any slips could get BOBOMIAO) Ignores 3 and expands (1 ± 2x)5 is 0/4 243, -810x, 1080x2 is full marks but243, -810, 1080 is GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. ak = 2. Then, x6 will appear in the term containing (2x) 6 and nowhere else. com Edexcel Internal Review 1 1. Solomon Press. But with the Binomial theorem, the process is relatively fast! Created by Sal Khan. And the sum $1-1+1-1+\cdots$ is not convergent, because the sequence of its partial sums (which is $1,0,1,0,1,0\cdots$) is not convergent (because it does not have a limit). This question is missing context or other details : Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. 3a. b Use your series expansion with a suitable value of x to obtain an estimate for 2. Show Step-by-step Solutions In the previous chapter (but not only), we also have explained how to expand the square and the cube of a binomial. 4: Binomial Theorem The binomial theorem provides a method of expanding binomials raised to powers without University of Glasgow - MSc Astronomy and Physics. AQA Core 2 5. (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3. c) Use the answer of part (b) to estimate, correct to 2 significant figures, the C2 Sequences & Series: Binomial Expansion PhysicsAndMathsTutor. The first two numbers in the Fibonacci sequence are 1, and each successive term is the sum of the previous two. Use Pascal’s triangle to quickly determine the binomial coefficients. 025-0. and . (2) (Total 6 marks) For the binomial expansion, in descending powers of x, of; 12 3 2. (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of (2 + kx)7. b. Feb 14, 2022 · Exercise 12. Therefore, the general term is an = 5n − 1. (12) 6. (4) (b) Use this expansion with your values of p and q together with an appropriate value of x C2 Sequences & Series: Binomial Expansion www. Familiarise with the formulae of a geometric sequence: nth term = ar^ (n - 1) and sum of first n terms = a (1 - r^n) / (1 - r) when. In the previous chapter (but not only), we also have explained how to expand the square and the cube of a binomial. Find the values of the constants a, b and c. (4+3x)5 ( 4 + 3 x) 5 Solution. £55 / hour. Pearson Education accepts no responsibility whatsoever for the accuracy or method of working in the answers given. Taking bookings for study leave, summer and 2024-2025. The exponents on b increase by one on each term going left to right. 1b 3 C2 2015 1 4. Nov 17, 2022 · This page titled 7: Sequences and Series, Mathematical Induction, and the Binomial Theorem is shared under a CC BY-NC-SA 3. The Binomial Series. y3 term of (y + 5)4. uk Edexcel Internal Review 1 1. Find the possible values of the constant a. The binomial theorem provides a method for expanding binomials raised to powers without directly multiplying each factor: (x + y)n = n ∑ k = 0(n k)xn − kyk. Hint: use Pascal’s triangle and binomial expansion: (a) x4 + 4x3 + 6x2 + 4x+ 1. Revision notes on 4. Three consecutive terms of an arithmetic series are a, b, and (3a + 4) respectively. C2 SEQUENCES AND SERIES Answers - Worksheet C 1 4a = 1 + 4x + 6x 2 + 4x3 + x b = 1 − 5x + 10x − 10x3 + 5x4 − x5 c = 1 + 3(4x) + 3(4x)2 + (4x)3 d = 1 + 3(−2y) + 3(−2y)2 + (−2y)3 = 1 + 12x + 48x2 + 64x3 = 1 − 6y + 12y2 − 8y3 e = 1 + 4(1 2 x) + 6(1 2 x)2 + 4(1 2 x)3 + (1 2 x)4 f = 1 + 3(1 3 y) + 3(1 3 y)2 + (1 3 y)3 = 1 + 2x + 3 2 To get an approximation you can consider a few terms from the expansion. All C3 Revsion Notes. (4) (Total 9 marks) (a) (i) Using the binomial expansion, or o therwise, express (2 + y)3 in the form Geometric Sequences and Series. --> 512 + 11520x + 115200x² --> 512 + 11520(0. Evaluate. 10C6 a4b6 = 10C4 a4b6 10 × 9 × 8 × 7 4 =. 3. 025) 8 giving your answer to 4 decimal places. com. C2 Sequences & Series: Binomial Expansion PhysicsAndMathsTutor. ) If we wish to expand an expression of the form , then we can use the above formula by replacing every with . a) Binomial Expansion 1 b) Binomial Expansion 2, c) Geometric Sequences 1 ,d) Geometric Sequences 2. Pascal’s Triangle. 2 Arithmetic Sequences and Series 409 8. This sequence is known as Pascal's triangle. x6 term of (x + 2)8. = 1 + 12x+ 48x2+ 64x3= 1 − 6y+ 12y2− 8y3. 4 in ascending powers of x up to and including the term in x3 is. (2) (Total 6 marks) 2. (a) Find the first 4 terms, in ascending powers of . 7 Feb 19, 2024 · The number of terms is n + 1. 43 JEE Main Mathematics Online (2019-2012) & Offline (2018-2002) Chapter-wise + Topic-wise Solved Papers 3rd Edition 2011-03-08 Brooks/Cole. (5) 4. 03 (2dp) Solomon Edexcel Worksheets and answers for the C2 module. 3 2 (2) (b) Find the first term of the sequence. Nov 16, 2022 · For problems 1 & 2 use the Binomial Theorem to expand the given function. 0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 1) Set the expansion expression equal to the new value. All C2 Revsion Notes. Give each term in its simplest form. Book Tutor. (3) (d) Find the sum to infinity of the sequence. Jan 2, 2012 · C2 Edexcel Core Mathematics January 2012 Question 3 Binomial Expansion 3. Example 1 Use the Binomial Theorem to expand (2x−3)4 ( 2 x − 3) 4. in the expansion. n + 1. (4) Given that the coefficient of . For instance, for "small" x, 1 + nx is a "reasonable" approximation for (1 + x)n. For example, 2+4+6+8+ is a series. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team. (4) Given that the coefficient of x2 in this expansion is 525, (b) find the possible values of a. (click to see video) One interesting example is the Fibonacci sequence. a . Term in x2 is 3 × 21 × (x a ) 2 = 6a2x2 C2 SEQUENCES AND SERIES Answers - Worksheet D page 4 Solomon Press 13 a = 312 + 12(311)(− 3 x) + 12 11 2 × (310)(− 3 x)2 + 12 11 10 32 ×× × (39)(− 3 x)3 + … = 531 441 − 708 588x + 433 026x2 − 160 380x3 + … b let 3 x = 0. Notice that this corresponds to picking the first two terms from the binomial theorem expansion (1 + x)n = 1 +(n1) x +(n2) x2 + ⋯ +xn. \displaystyle {\left ( {a}+ {b}\right)}^ {5} (a +b)5. [4] (iii) Hence find the coefficient of . Find the first 4 terms, in ascending powers of x, of the binomial expansion of. Next use the formula to determine the 50th partial sum of the given arithmetic sequence. Nov 16, 2022 · This is useful for expanding (a+b)n ( a + b) n for large n n when straight forward multiplication wouldn’t be easy to do. This topic is included in Paper 1 for AS-level Edexcel Maths and Papers 1 & 2 for A-level Edexcel Maths. Questions and model answers on 4. (2) (c) Find the sum of the first 15 terms of the sequence. 001)² --> 523. --> (2+5x)⁹ = 2. University of Bristol - MEng Mechanical and Electrical Engineering. 5. (4) (c) Hence find the coefficient of x in the expansion of . 4 3 + x(1 12 ) in ascending powers of up to and including the x term in x3 is 1 + 9x + px2 + qx3, 12x < 1. Here is a set of practice problems to SEQUENCES AND SERIES Answers - Worksheet C. (a) Find the first 4 terms, in ascending powers of x, of the binomial… C2-Sequences-Series-C-Simple-Binomial-Expansion-Answers. Find the first 3 terms, in ascending powers of x, of the binomial expansion Jun 20, 2020 · C2 Sequences Series: Binomial Expansion Edexcel Internal Review 1 1 a Find the first 4 terms in ascending powers of x of the binomial expansion of 1 + ax7 where a is a constant&#8230; May 27, 2024 · Get Sequences and Series Multiple Choice Questions (MCQ Quiz) with answers and detailed solutions. [4] C2 Sequences and Series. where b is a non-zero constant. 1b 4 C2 June 2014R 1 4. 002 ∴ x = 0. Symbolise this as a, ar, ar²,…. 0. (a + b) 2 = a 2 + 2ab + b 2. Figure 12. c= 1 + 3(4x) + 3(4x)2+ (4x)3d= 1 + 3(−2y) + 3(−2y)2+ (−2y)3. The binomial expansion of (1 + 12 x ) 3. 3 Geometric Sequences and Series 418 8. xa rt jh fo br dg tb vc wq mt