NYJC 2026 FM TP - FM Recurrence Relations

NANYANG JUNIOR COLLEGE DEPARTMENT OF MATHEMATICS 2026 J2 H2 FM Recurrence Relations 1. To improve the biodiversity in the forest of Anteeyew, a botanist plans to introduce a new species of wildflower to the forest. The wildflower produces seeds which will germinate at different times. It is known that of the seeds produced by each wildflower each week, on average, 3 seeds will germinate in one week’s time, and 5 seeds will germinate in two weeks’ time. Let nW denote the number of wildflowers present in the forest of Anteeyew at the end of the thn week since the introduction of the wildflowers to the forest. (a) Show that 1245n n nW W W −−=+ for 2n . [1] Initially, the botanist plants 10 wildflowers in the forest of Anteeyew. (b) Taking 0 10W = , state the value of 1W . Hence, find the general formula for nW for 0n . [4] (c) Give one possible reason for why the recurrence relation in part (a) may not be a suitable model for the population of wildflower s in the forest of Anteeyew. [1] A new insect species is attracted to the wildflowers in the forest of Anteeyew. Let nI denote the population of the insect at the end of the thn week since the introduction of the wildflowers to the forest. With the presence of this new insect species, the populations of wildflowers and insects due to their interactions may be modelled by 11 10 , 0 , 1 n nn nW W I n−− == + and 11 5 , 0 2 5 , 1 n nn nI I W n−− == + −  . (d) Obtain a second order recurrence relation for nI and show that the insect population approaches ( )( )2 1 2 n ab++ in the long run, where a and b are rational constants to be found. [6] 2. A sequence 0 1 2, , ,u u u  is given by 21 8 15n n nu u u++ =− , 01 1uu== . (a) Show that ( )2 1 13 5 3n n n nu u u u+ + +− = − . [1] (b) Let 1 3n n nv u u +=− . Write down a first-order recurrence relation for nv , and hence express nv in terms of n. [3] (c) Let 1 5n n nw u u +=− . Write down a first -order recurrence relation for nw , and hence show that ( )43 n nw =− . [3] (d) By finding and solving a pair of simultaneous linear equations in 1nu + and nu , or otherwise, express nu in terms of n for 0n . [2]

3 The aggregate expenditure of Country X is given by the sum of household consumption and government expenditure in a year. Let and nnYC be the aggregate expenditure and household consumption at the end of the nth year respectively, where 1.n Let G be the government expenditure which is assumed to be constant from year to year. In an economic model, the household consumption in each year is assumed to be a linear function of the aggregate expenditure of the preceding year. (a) Based on the information above, find a recurrence relation for the aggregate expenditure in the form 1 f ( ).nnYY+ = Hence, show that ( )11 1 1 n n n mY G k m Y m +  −= + +  − for all positive integer n, where k and m are constants. [4] (b) If 0 1,m find the long term aggregate expenditure of the country, in terms of , and .G k m [2] 4. The terms in the sequence 1u , 2u , 3u … satisfy the recurrence relation 2 12 2 2 4 ,n n nu u u  −−=− where  is a positive constant. (a) Find the general solution of this recurrence relation. [4] (b) Find the set of values of  for which the sequence is convergent. [2]