Sequences

To continue a sequence, find what happens from one term to the next. Subtract each term from the one after it. If the difference is the same every time, keep adding it. If the differences themselves change, look for a pattern in them. For 1, 3, 6, 10, 15 the differences are 2, 3, 4, 5, so the next two terms are 21 and 28.

In Extended work, Tn means the nth term of sequence T. So T₃ is the 3rd term. Write the new terms in order.

The term-to-term rule tells you how to get from one term to the next. Write it in words, with an operation and a number: add 7, subtract 6, divide by 10. It is not the nth term: it has no n in it.

To find the rule, test the difference first. If the difference changes but each term is the same multiple of the one before, the rule is to multiply or divide.

Two sequences can be linked term by term. If Cn = An + Bn, each term of C is the term of A plus the term of B in the same position.

Rearrange as you would an equation: An = Cn - Bn. Find the nth term of the sequence you know, then subtract. Take care with brackets when you subtract an expression.

In a linear sequence the terms go up or down by the same amount, the common difference d. The nth term is dn + c.

To find c, go back one step from the first term: c = first term - d. Check by putting n = 1 and n = 2 into your answer.

For a decreasing sequence d is negative. Once you have the nth term, set it equal to a number and solve to test whether that number is in the sequence. It is a term only if n is a whole number.

To find a term, put its position in place of n. For the first three terms use n = 1, n = 2 and n = 3. The first term is n = 1, not n = 0.

The nth term means the term in position n. It is not the 9th term. A formula such as 3n - 1 is the nth term, and a term is in the sequence if some whole n gives it.

A simple quadratic sequence compares with the square numbers 1, 4, 9, 16, 25. The simplest have nth term n² plus or minus a number. A simple cubic sequence compares with the cube numbers 1, 8, 27, 64. The simplest have nth term n³ plus or minus a number.

Write the square or cube numbers under the sequence and see what you must add or subtract. The differences between the terms are not constant, so a linear nth term does not fit. To use the nth term, square or cube n first: 1² = 1, not 2.

In an exponential sequence each term is the previous term multiplied by the same number. The term-to-term rule is multiply by that number.

The nth term has the form a × rⁿ⁻¹, where a is the first term and r is the number you multiply by. Do not write r × n: that is a linear sequence. Leave 4 × 3ⁿ⁻¹ as it is: it is not 12ⁿ⁻¹. Check your answer with n = 1 and n = 2.