Sequences & Series

Edexcel IAL WMA12/01/P2/Oct 2019/Q2 (Geometric Series)

The adult population of a town at the start of the start of 2019 in 25000

A model predicts that the adults population will increase by 2% each year, so that the number of adults in the population at the start of each of each year following 2019 will form a geometric sequence.

• Find, according to the model, the adult population of the town at the start of 2032

(3)

It is also modelled that every member of the adult population gives £5 to local charity at the start of each year.

• Find, according to these models, the total amount of money that would be given to local charity by the adult population of the town from 2019 to 2032 inclusive. Given your answer to the nearest £1000

(3)

SOLUTION​​ 

a- ​​ We need to find the population at the start of 2032, which is 14 years after 2019.​​ 

Using the formula of gemetric sequence, where​​ a=25, 000,​​ r=1.02, and​​ n=14.​​ 

Un=arn-1

U14=25, 000 x 1.0213

U14=32, 340.167 . .. . 

U14=32,300

b- ​​ We are asked to find the sum of money given in charity by the adults during the perios 2019 to 2032. So, first we will find the sum of the adult population in the peroid 2019 to 2032 (14 years). Then, we will multiply £5 with the population to find the total amount given in charity.​​ 

So, first finding​​ S14, where​​ a=25, 000,​​ r=1.02, and​​ n=14.

Since​​ r>1, we will use the second equation of the flash card.​​ 

Sn=arn-1r-1

S14= 25, 000 1.0214-11.02-1

S14=399348.4538 

The amount we will have in the charity during this 14 years period is​​ 5×S14

Amount=5×S14

=5×399348.

= £1, 996,700 

The answer needs to be in nearest £1000, so the final answer is​​ £1, 997,000.​​ 

Amount=£1, 997,000