12 Further mathematics Rules for the nth term in a sequence modelling

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

Rules for the nth term in a sequence modelling geometric growth or decay While we can generate as many terms as we like in a sequence using a recurrence relation for geometric 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 geometric growth or decay We invest $2000 in a compound interest investment paying 5% interest per annum, compounding yearly. 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 = 1.05Vn Using this recurrence relation we can write out the sequence of terms generated as follows: V0 = 2000 V1 = 1.05V0 V2 = 1.05V1 = 1.05(1.05V0) = 1.052V0 V3 = 1.05V2 = 1.05(1.052V0) = 1.053V0 V4 = 1.05V3 = 1.05(1.053V0) = 1.054V0 and so on.

Rules for the nth term in a sequence modelling geometric growth or decay Following this pattern, after n year’s interest has been added, we can write: Vn = 1.05n x V0 With this rule, we can now predict the value of the nth term in the sequence without having to generate all of the other terms first.

Rules for the nth term in a sequence modelling geometric growth or decay For example, using this rule, the value of the investment after 20 years would be: V20 = 1.0520 × 2000 = $5306.60 (to the nearest cent) This rule can be readily generalised to apply to any geometric growth or decay situation.

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

Rules for the nth term in a sequence modelling geometric growth or decay Exactly the same rule will work for both growth and decay because growth or decay depends on the value of R, not the format of the calculation. This general rule can also be applied to compound interest loans and investment and reducing-balance depreciation.

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

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

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

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

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

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

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

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

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