Sequences and Series - The Binomial Theorem
Sequences and Series - The Binomial Theorem
Sequences and Series
Arithmetic Sequences and Series
- Understand the concept of a sequence as a list of numbers where each term has a fixed rule for progression.
- Define arithmetic sequence as a sequence in which each term after the first is found by adding a constant difference to the preceding term. Symbolise this as a, a+d, a+2d,…
- Comprehend the formulae applicable to an arithmetic sequence: nth term = a + (n - 1)d and sum of first n terms = 0.5n [2a + (n - 1)d].
- Recognise how to find the common difference, first term, and nth term of an arithmetic sequence.
Geometric Sequences and Series
- Define geometric sequence as a sequence in which each term after the first is found by multiplying the preceding term by a constant ratio. Symbolise this as a, ar, ar²,…
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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 r < 1. - Understand the concept of a geometric series as the sum of the terms in a geometric sequence.
- Determine the common ratio, first term, and nth term of a geometric sequence.
The Binomial Theorem
- Comprehend the Binomial Theorem, which states that (a + b)^n = ∑^n_(k=0) nCk * a^(n-k) * b^k, where nCk denotes the combinations of n items taken k at a time.
- Apply the Binomial Theorem to expand and simplify binomial expressions for positive integer exponents.
- Be capable of explaining the concept of Binomial Coefficients, which are the coefficients of the terms in a binomial expansion, given by the formula nCk = n! / [k!(n - k)!].
- Understand how to use Pascal’s Triangle to find binomial coefficients for binomial expansions.
- Interpret term position in a binomial expansion: the ‘r+1’th term has an exponent of ‘n-(r-1)’ for the first term in the binomial expression.
- Apply the general term in the binomial expression to find specific terms without fully expanding the expression.
- Develop knowledge of applying the binomial theorem in real-world situations, such as probability calculations.