Grade 11 Sequences, Series & Matrices Practice — Easy

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Question 1 of 28: In an arithmetic sequence, consecutive terms differ by a constant called:

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Answer key for parents & teachers (28 questions)
  1. In an arithmetic sequence, consecutive terms differ by a constant called: — The common difference d. Arithmetic sequences have a constant common difference d between consecutive terms.
  2. In a geometric sequence, consecutive terms have a constant: — Ratio r. Geometric sequences have a constant common ratio r = aₙ₊₁/aₙ.
  3. The nth term of arithmetic sequence: aₙ = ? — a₁ + (n−1)d. aₙ = a₁ + (n−1)d. Start at a₁, add d a total of (n−1) times.
  4. The nth term of a geometric sequence: aₙ = ? — a₁·rⁿ⁻¹. aₙ = a₁·rⁿ⁻¹. Start at a₁, multiply by r a total of (n−1) times.
  5. Arithmetic sequence: 3, 7, 11, 15, … The 10th term is: — 39. a₁ = 3, d = 4. a₁₀ = 3 + 9(4) = 3 + 36 = 39.
  6. Geometric sequence: 2, 6, 18, 54, … The common ratio r is: — 3. r = 6/2 = 3 (or 18/6 = 3).
  7. Identify: 1, 1, 2, 3, 5, 8, 13, … This is: — Fibonacci. The Fibonacci sequence: each term is the sum of the two preceding terms. Not arithmetic or geometric.
  8. The Binomial Theorem expands (a+b)ⁿ as: — Σ C(n,k)·aⁿ⁻ᵏ·bᵏ for k=0 to n. (a+b)ⁿ = Σₖ₌₀ⁿ C(n,k)·aⁿ⁻ᵏ·bᵏ. Each term combines a binomial coefficient with powers of a and b.
  9. The binomial coefficient C(n,k) appears in row __ of Pascal's Triangle: — n. Row n of Pascal's Triangle gives the coefficients C(n,0), C(n,1), …, C(n,n) for the expansion of (a+b)ⁿ.
  10. How many terms are in the expansion of (a+b)ⁿ? — n+1. k goes from 0 to n, giving n+1 terms.
  11. Expand (x+1)² using the Binomial Theorem: — x² + 2x + 1. (x+1)² = C(2,0)x² + C(2,1)x + C(2,2) = x² + 2x + 1.
  12. (x+y)³ expanded equals: — x³ + 3x²y + 3xy² + y³. (x+y)³ = x³ + 3x²y + 3xy² + y³. Row 3 of Pascal's Triangle: 1, 3, 3, 1.
  13. What is C(6,0)? — 1. C(6,0) = 6!/(0!·6!) = 1. This is the first entry in every row of Pascal's Triangle.
  14. In Pascal's Triangle, each entry is: — The sum of the two entries above it. Pascal's rule: C(n,k) = C(n−1,k−1) + C(n−1,k). Each entry is the sum of the two above it.
  15. Expand (1 + x)³: — 1 + 3x + 3x² + x³. (1+x)³ = C(3,0) + C(3,1)x + C(3,2)x² + C(3,3)x³ = 1 + 3x + 3x² + x³.
  16. (a + b)⁰ = ? — 1. Any nonzero expression to power 0 equals 1. (a+b)⁰ = 1. This is row 0 of Pascal's Triangle.
  17. What is the first term (k=0) in the expansion of (x+3)⁵? — x⁵. k=0: C(5,0)·x⁵·3⁰ = 1·x⁵·1 = x⁵.
  18. What is the last term (k=n) in the expansion of (x+3)⁵? — 3⁵ = 243. k=5 (=n): C(5,5)·x⁰·3⁵ = 1·1·243 = 243.
  19. A 3×4 matrix has: — 3 rows and 4 columns. An m×n matrix has m rows and n columns. A 3×4 matrix has 3 rows and 4 columns.
  20. Two matrices can be added only if they have: — The same dimensions (m×n). Matrix addition requires identical dimensions — same number of rows AND columns.
  21. Matrix addition is performed: — Element by element. Add corresponding entries: (A+B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ.
  22. If A is 2×3 and B is 3×4, then AB is: — 2×4. A(2×3) × B(3×4): inner dimensions (3=3) match. Result is 2×4.
  23. The determinant of [[a,b],[c,d]] is: — ad − bc. det([[a,b],[c,d]]) = ad − bc. This "cross multiply and subtract" rule applies to 2×2 matrices.
  24. det([[5, 2], [3, 4]]) = ? — 14. det = (5)(4) − (2)(3) = 20 − 6 = 14.
  25. A · A⁻¹ = ? — I (identity matrix). By definition, A · A⁻¹ = I (the identity matrix). This is the defining property of the inverse.
  26. The 2×2 identity matrix is: — [[1,0],[0,1]]. Identity I = [[1,0],[0,1]]. Multiplying any 2×2 matrix by I returns the original matrix.
  27. det([[2,0],[0,3]]) = ? — 6. det = (2)(3) − (0)(0) = 6. Diagonal matrices have det = product of diagonal entries.
  28. Scalar multiplication: 3 · [[1,2],[3,4]] = ? — [[3,6],[9,12]]. Multiply every entry by 3: [[3,6],[9,12]].