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Use a logarithmic fixed-point graph to compare terms

The diagram shows y=ln⁡(x+2)−x/5y=\ln(x+2)-x/5. Its positive root is α\alpha. Find α\alpha. Positive terms satisfy xn+1=5ln⁡(xn+2)x_{n+1}=5\ln(x_n+2).

(i) Prove algebraically that convergence implies limit α\alpha.

(ii) Use a calculator to describe the behaviour for x1=1x_1=1 and x1=20x_1=20.

(iii) By considering xn+1−xnx_{n+1}-x_n, prove increase when 0<xn<α0<x_n<\alpha and decrease when xn>αx_n>\alpha.

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Graph h(x)=ln(x+2) minus x/5, with vertical asymptote x=minus 2 and positive root alpha approximately 13.8001. It is positive for 0<x<alpha and negative for x>alpha.
Graph h(x)=ln(x+2) minus x/5, with vertical asymptote x=minus 2 and positive root alpha approximately 13.8001. It is positive for 0<x<alpha and negative for x>alpha.
Graph h(x)=ln(x+2) minus x/5, with vertical asymptote x=minus 2 and positive root alpha approximately 13.8001. It is positive for 0<x<alpha and negative for x>alpha.

[2008/IJC/MYE/12(modified)]

CJC · Discussion · Question 15

The sign graph of h gives the sign of the increment after multiplying by 5.

Read the worked solution