Question 1

Paper 1, no calculatorHard6 marks

Let m∈Z+m\in\mathbb Z^+ Figure 1 shows a rectangle with side lengths m+1m+1 and m+2m+2 divided into four smaller rectangles.

It is claimed that integers AA and BB can be chosen so that, for every m∈Z+m\in\mathbb Z^+

1m+1+1m+2+Am+Bm2+3m+2=3m+1.\frac{1}{m+1}+\frac{1}{m+2}+\frac{Am+B}{m^2+3m+2} =\frac{3}{m+1}.
2m2m2mm+1m12mm+2
(a)

Using the areas shown in Figure 1, write an identity relating the area of the whole rectangle to the areas of the four smaller rectangles. Use ≡\equiv to denote an identity.

[1]
(b)

If the claim is true, the equality for m=1m=1 requires a particular relation between AA and BB Determine this relation.

[1]
(c)

Determine AA and BB so that the claim is an identity for every m∈Z+m\in\mathbb Z^+ Then, starting with the left-hand side, prove the resulting identity. Use your answer to part (b).

[4]

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