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.

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f'(x) = 3sin(3x).
f(1.6) = 1 − cos(3 × 1.6) = 0.931.
f'(1.6) = 3sin(3 × 1.6) = −2.988.
Using the Newton-Raphson formula, x₁ = 1.6 − 0.931/−2.988 = 1.912.

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f'(x) = 5cos(5x) − 2.
f(0.3) = sin(5 × 0.3) − 2(0.3) = 0.397.
f'(0.3) = 5cos(5 × 0.3) − 2 = −1.646.
Using the Newton-Raphson formula, x₁ = 0.3 − 0.397/−1.646 = 0.541.

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Generating the first few terms, 1, −2, 1, −2, …
So, the first 50 terms have twenty-five 1 and twenty-five −2.
So, sum = 25(1) + 25(−2) = −25.

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u₂ = 100 − 18 = 82.
u₃ = 100 − 82 = 18.
u₄ = 100 − 18 = 82.

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42446 = u₂² + 10, so u₂ = √42436 = 206.
206 = u₁² + 10, so u₁ = √196 = 14.