Binomial Theorem

The binomial theorem gives a quick way to expand powers of a binomial such as (a + b)^n without repeated multiplication. The chapter begins with history, including Pascal's triangle, called Meru Prastara by the ancient Indian mathematician Pingala. Each row of the triangle begins and ends with 1, and every other entry is the sum of the two entries above it. These entries are the combinations nC0, nC1, ..., nCn. You will learn the statement and proof of the binomial theorem for positive integral indices: (a + b)^n = nC0 a^n + nC1 a^(n-1) b + ... + nCn b^n. It has n + 1 terms, the powers of a decrease while those of b increase, and the sum of the indices in each term is n. The chapter covers special cases such as (1 + x)^n and (a - b)^n, the fact that the sum of the coefficients is 2^n, and simple applications such as finding 99^5 or (0.99)^5, comparing numbers like 1.1^10000 and 1000, and proving divisibility results.

Detailed Worksheet: Binomial Theorem Section A - Definitions (10 marks) 1. State the binomial theorem for a positive integral index n. (2 marks) 2. Write the first five rows of Pascal's triangle. (2 marks) 3. How many terms are there in the expansion of (a + b)^n? What is the sum of the indices of a and b in each term? (2 marks) 4. Write the expansion of (1 + x)^4. (2 marks) 5. Show that the sum of the binomial coefficients in the expansion of (1 + x)^n is 2^n. (2 marks) Section B - Calculations and Applications (15 marks) 6. Expand (2x - 3)^4 using the binomial theorem. (3 marks) 7. Using the binomial theorem, evaluate 102^5. (3 marks) 8. Using the binomial theorem, find (0.99)^5 correct to three decimal places. (3 marks) 9. Find (a + b)^4 - (a - b)^4 and hence evaluate (sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4. (3 marks) 10. Using the binomial theorem, show that 9^(n+1) - 8n - 9 is divisible by 64 for every positive integer n. (3 marks) Section C - Diagrams (10 marks) 11. Draw Pascal's triangle up to the row for n = 6 and show how each entry is formed from the row above. (4 marks) 12. Using Pascal's triangle, write the expansion of (x + y)^6 and mark the symmetry of the coefficients. (3 marks) 13. Draw a diagram to show the expansion of (a + b)^2 and (a + b)^3 as areas and volumes of squares and cubes. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Prove the binomial theorem for positive integral index n by mathematical induction or by a combinatorial argument. (5 marks) 15. Using the binomial theorem, show that 1.1^10000 > 1000. Explain why only the first two terms are needed. (5 marks) 16. Expand (x + 1/x)^6 and find the term independent of x by inspection. Also expand (x^2 + 3/x)^4. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show all steps. Use nCr notation for combinations.