Sequences and Series

Sequences and Series - The Binomial Theorem

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,…
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.

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.