Sequences Revision Notes

A sequence is an ordered list of numbers arranged according to a specific rule or pattern. Each number in the sequence is called a term, and the position of a term is indicated by a positive integer n, starting from 1 for the first term.

Subscript notation is commonly used to represent terms in a sequence. For example, T_n denotes the nth term of sequence T. This notation helps to express general formulas and refer to any term without listing all previous terms.

• Sequence: ordered list of numbers following a pattern.
• Term: an individual number in the sequence.
• Position n: the place of a term in the sequence, where n = 1, 2, 3,...
• Subscript notation: T_n means the term at position n in sequence T.

Recognising the pattern in a sequence is essential to finding its nth term. Common types of patterns include: linear, quadratic, cubic, and exponential.

Linear sequences increase or decrease by a constant difference between terms.

Quadratic sequences have differences between terms that change linearly, often related to n squared terms in the formula for T_n. The second differences between terms are constant in quadratic sequences, which helps identify them reliably through calculation of differences between consecutive terms and then differences of those differences (second differences). For example, the sequence 2, 6, 12, 20, 30 has first differences 4, 6, 8, 10 and second differences 2, 2, 2, indicating a quadratic pattern. Cubic sequences involve terms related to n cubed, with constant third differences between terms. Exponential sequences multiply by a constant factor each time, such as 2, 4, 8, 16, where each term is multiplied by 2 to get the next term.

• Linear: constant first difference.
• Quadratic: constant second difference.
• Cubic: constant third difference.
• Exponential: constant ratio between terms.

The nth term formula expresses the value of the term at position n without listing all previous terms. The form of the formula depends on the pattern type.

For linear sequences, the nth term is generally T_n = an + b, where a is the common difference and b is the first term adjusted for n=1.

For quadratic sequences, the nth term is of the form T_n = an^2 + bn + c. To find a, b, and c, use the first three terms and solve simultaneous equations or use the method of differences to find a = (second difference)/2, then find b and c accordingly. Cubic sequences have nth terms of the form T_n = an^3 + bn^2 + cn + d, requiring more terms and solving systems of equations to find coefficients. Exponential sequences have nth terms of the form T_n = ar^{n-1}, where a is the first term and r is the common ratio between terms.

• Linear nth term: T_n = an + b.
• Quadratic nth term: T_n = an^2 + bn + c.
• Cubic nth term: T_n = an^3 + bn^2 + cn + d.
• Exponential nth term: T_n = ar^{n-1}.

Once the nth term formula is known, it can be used to find any term in the sequence directly, without listing all previous terms. This is especially useful for large values of n.

For example, consider the linear sequence 3, 7, 11, 15,... The common difference is 4, so the nth term is T_n = 4n - 1 (since T_1 = 3 = 4(1) - 1). To find the 10th term, substitute n = 10: T_{10} = 4(10) - 1 = 39.

Similarly, for the quadratic sequence 2, 6, 12, 20,... with nth term T_n = n^2 + n, the 5th term is T_5 = 5^2 + 5 = 25 + 5 = 30.

• Use nth term formula to find any term directly.
• Substitute the desired term position n into the formula.
• Check results by comparing with known terms if possible.

Subscript notation clarifies which term is being referred to and is essential when working with sequences algebraically.

For example, T_1 is the first term, T_2 the second, and so on. When writing formulas or solving problems, using T_n helps generalise the sequence and communicate clearly.

In problems involving multiple sequences, different letters with subscripts distinguish terms, such as A_n and B_n for sequences A and B respectively.

• T_n denotes the nth term of sequence T.
• Subscripts help generalise and communicate terms.
• Use different letters for different sequences with subscripts.