Rules for the nth term in a sequence modelling linear growth or decay

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Rules for the nth term in a sequence modelling linear growth or decay 12 Further mathematics Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay While we can generate as many terms as we like in a sequence using a recurrence relation for linear growth and decay, it is possible to derive a rule for calculating any term in the sequence directly. This is most easily seen by working with a specific example.

Rules for the nth term in a sequence modelling linear growth or decay For instance, if you invest $2000 in a simple interest investment paying 5% interest per annum, your investment will increase by the same amount, $100, each year. If we let Vn be the value of the investment after n years, we can use the following recurrence relation to model this investment: V0 = 2000, Vn+1 = Vn + 100

Rules for the nth term in a sequence modelling linear growth or decay Using this recurrence relation we can write out the sequence of terms generated as follows: V0 = 2000 = V0 + 0 × 100 (no interest paid yet) V1 = V0 + 100 = V0 + 1 × 100 (after 1 years’ interest paid) V2 = V1 + 100 = (V0 + 100) + 100 = V0 + 2 × 100 (after 2 years’ interest paid) V3 = V2 + 100 = (V0 + 2 × 100) + 100 = V0 + 3 × 100 (after 3 years’ interest paid) V4 = V3 + 100 = (V0 + 3 × 100) + 100 = V0 + 4 × 100 (after 4 years’ interest paid) and so on.

Rules for the nth term in a sequence modelling linear growth or decay Following this pattern, after n years’ interest has been added, we can write: Vn = 2000 + (n × 100) With this rule, we can now determine the value of the nth term in the sequence without having to generate all of the other terms first. For example, using this rule, the value of the investment after 20 years would be: V20 = 2000 + (20 × 100) = $4000 This rule can be readily generalised to apply to any linear growth or decay situation.

Rules for the nth term in a sequence modelling linear growth or decay For a recurrence rule of the form: V0 = initial value, Vn+1 = Vn + D where (D constant) the value of the nth term generated by this recurrence relation is: Vn = V0 + (n x D) for linear growth. Vn = V0 − (n x D) for linear decay. This general rule can then be applied to simple interest investments and loans, flat-rate depreciation and unit-cost depreciation.

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Rules for the nth term in a sequence modelling linear growth or decay

Exercise 8D – All Even Questions WORK TO BE COMPLETED Exercise 8D – All Even Questions