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AQA A-level Maths Recurrence relations 1. Recurrence relationships
2. Graphical representation of Netwon-Raphson method
3. Newton-Raphson method
4. Newton-Raphson method problems in context

A recurrence relationship can be used to generate all the terms of a sequence. It describes each term as a function of the previous term. The Newton-Raphson method uses tangent lines to find approximations of roots of equations in the form f(x) = 0. A value for x₀ is chosen and a tangent at that point is drawn. The next x value is taken to be the point at which the tangent intersects the x-axis. This process is continued to find increasingly accurate approximations of the root. The Newton-Raphson formula can be used to calculate increasingly accurate approximations of a root, given a starting value. If the starting value is chosen to be a turning point, the formula cannot be used, since its derivative will be 0 and division by 0 in the formula will not be possible. If the starting value is chosen to be near a turning point, the gradient will be small, therefore the tangent will intersect the x-axis a long way away from the starting value, and therefore the Newton-Raphson method may fail. The Newton-Raphson method can be used to model situations and find their solutions. 1

Using x = 1.6 as the first approximation, calculate a second approximation for the given root of the graph. 2

Using x = 0.3 as a first approximation, calculate a second approximation for the root of the given function. 3

The recurrence relationship of a sequence is given by u_(n + 1) = u_n² − 3, where u₁ = 1. Calculate the sum of the first 50 terms. 4

The recurrence relationship of a sequence is given by u_(n+1) = 100 − u_n, where u₁ = 18. Calculate the next three terms of the sequence. 5

Given the recurrence relationship u_(n + 1) = u_n² + 10, fill in the blanks of the sequence.

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