Exercise 3a81P1^
3
ExamPlE 3.5 Jamila starts a part-time job on a salary of $9000 per year, and this increases by
an annual increment of $1000. Assuming that, apart from the increment, Jamila’s
salary does not increase, find
(i) her salary in the 12th year
(ii) the length of time she has been working when her total earnings are $100 000.
SOlUTION
Jamila’s annual salaries (in dollars) form the arithmetic sequence
9000, 10 000, 11 000, ....
with first term a = 9000, and common difference d = 1000.
(i) Her salary in the 12th year is calculated using:
uk = a + (k − 1)d
⇒ u 12 = 9000 + (12 − 1) × 1000
= 20 000.
(ii) The number of years that have elapsed when her total earnings are $100 000
is given by:(^) Sn=+^12 [ 21 an(– )d]
where S = 100 000, a = 9000 and d = 1000.
This gives 100 000 = 21 n[ 2 ×+ 9000 1000 (–n 1 )].
This simplifies to the quadratic equation:
n^2 + 17 n − 200 = 0.
Factorising,
(n − 8)(n + 25) = 0
⇒ n = 8 or n = −25.
The root n = −25 is irrelevant, so the answer is n = 8.
Jamila has earned a total of $100 000 after eight years.
ExERCISE 3a 1 Are the following sequences arithmetic?
If so, state the common difference and the seventh term.
(i) 27, 29, 31, 33, ... (ii) 1, 2, 3, 5, 8, ... (iii) 2, 4, 8, 16, ...
(iv) 3, 7, 11, 15, ... (v) 8, 6, 4, 2, ...
2 The first term of an arithmetic sequence is −8 and the common difference is 3.
(i) Find the seventh term of the sequence.
(ii) The last term of the sequence is 100.
How many terms are there in the sequence?