Question

Shelim starts his new job on a salary of £14 000. He will receive a rise of £1500 a year for each full  year that he works, so that he will have a salary of £15 500 in year 2, a salary of £17 000 in year 3  and so on. When Shelim’s salary reaches £26 000, he will receive no more rises. His salary will  remain at £26 000.

a.   Show that Shelim will have a salary of £26 000 in year 9.

b.   Find the total amount that Shelim will earn in his job in the first 9 years.

Anna starts her new job at the same time as Shelim on a salary of £A. She receives a rise of £1000   year for each full year that she works, so that she has a salary of £(A+1000) in year 2, £(A+2000)  in year 3 and so on. The maximum salary for her job, which is reached in year 10, is also £26 000. 

c.   Find the difference in the total amount earned by Shelim and Anna in the first 10 years.

Solution

a.
 

It is evident that the salaries received by Shelim form an arithmetic sequence.

From the given information we can collect following data about the said arithmetic sequence.

Expression for difference  in Arithmetic Progression (A.P) is:

Therefore;

We are required to show that salary for  9^th year will be £26 000.

It is evident that we are looking for 9^th term of above given arithmetic sequence.

Expression for the general term  in the Arithmetic Progression (A.P) is:

Therefore;

b.
 

We are required to find the total amount earned during 9 years of job by Shelim.

It is evident that we are looking for the sum of 9 terms of above given arithmetic sequence.

Expression for the sum of  number of terms in the Arithmetic Progression (A.P) is:

Therefore;

Hence, his total earning during 9 years will be £180000.

c.
 

It is evident that the salaries received by Anna form an arithmetic sequence.

From the given information we can collect following data about the said arithmetic sequence.

Expression for difference  in Arithmetic Progression (A.P) is:

Therefore;

First we need to find the total amount earned by Anna during 10 years of job.

It is evident that we are looking for the sum of 10 terms of above given arithmetic sequence.

Expression for the sum of  number of terms in the Arithmetic Progression (A.P) is:

However, first we need to find first salary/term.

Expression for the general term  in the Arithmetic Progression (A.P) is:

Now we can find sum of all salaries received in ten years.

Hence, his total earning during 10 years will be £215000.

Since there is no increment in Shelim’s salary after year 9, therefore, his total earning in 10 years  will be sum 9 year’s earning as found in (b) plus salary of 10th year.