Sequences & Series

Edexcel IAL WMA12/01/P2/June 2019/Q7 (Arithmetic & Geometric Series)

Kim starts working for a company.

• In year 1 her annual salary will be £16200

• In year 10 her annual salary is predicted to be £31 500

Model A assumes that her annual salary will increase by the same amount each year.

• According to model A, determine Kim's annual salary in year 2.

(3)

Model B assumes that her annual salary will increase by the same percentage each year.

• According to model B, determine Kim's annual salary in year 2. Give your answer to the nearest £10

(3)

• Calculate, according to the two models, the difference between the total amounts that Kim is predicted to earn from year 1 to year 10 inclusive. Give your answer to the nearest £10

(3)

SOLUTION​​ 

a- Since Kim’s salary increases each year by the same amount in model A, it is an arithmetic sequence. Let us first find the value of d (the common difference).​​ 

  Un=a+n-1 d

Where,​​ Un=31500,​​ a=16,200, and​​ n=10.

  U10=16200+10-1 d

31,500=16,200+9d 

d=31,500-16,2009

d=1700

So in year 2 according to plan A.​​ 

  Un=a+n-1 d

U2=16,200+17000

U2= £17,900 

b- Since Kim’s salary increases each year by the same percentage in model B, it is geometric sequence. Let us first find the value of r (the common ratio).​​ 

Un=arn-1

Where,​​ u10=31500,​​ a=16,200, and​​ n=10.

U10=16200×r10-1

31500=16200 x r9

r= 31500162009 = 3151629 

r=1.07668 . . 

So in year 2 according to plan B.​​ 

U2=ar2-1=ar

U2=16200 x 3151629

U2=17442.29

U2=  £17, 440

c- Finding the difference in the​​ Sn​​ of both the plans.​​ 

In plan A, the sum at the end of 10 years would be

Sn=n2a+l

S10=10216200+31500

S10=238, 500

In plan B, the sum at the end of 10 years would be

Sn=arn-1r-1

S10=16,2001.07710-11.077-1

S!0=231, 019.244

Hence, the difference will be​​ 

Diff=238, 500-231,019.244 . . .

=7480.76 

In the nearest £10

=£7480