Sequences & Series

Edexcel IAL WMA12/01/P2/June 2021/Q1 (Arithmetic Series)

Adina is saving money to buy a new computer. She saves £5 in week 1, £5.25 in week 2, £5.50 in week 3 and so on until she has enough money, in total, to buy the computer.​​ 

She decides to model her savings using either an arithmetic series or a geometric series.​​ 

Using the information given,​​ 

• ​​ 

• state with a reason whether an arithmetic series or a geometric series should be used,​​ 

• write down an expression, in terms of n, for the amount, in pounds (£), saved in week n.​​ 

(3)

Given that the computer Adina wants to buy costs £350​​ 

• find the number of weeks it will take for Adina to save enough money to buy the computer.​​ 

(4)

SOLUTION​​ 

a-i- ​​ Let see first for geometric sequence, ​​ 

r=U2U1=U3U2

U2U1=5.255=2120

U3U2=5.505.25=2221

Since​​ U2U1≠U3U2, ​​ she can’t use geometric sequence.​​ 

Lets try now, the arithmetic sequence.​​ 

d=U3-U2=5.50-525= 0.25

d=U2-U1=5.25-5.00=0.25

d=0.25

Hence, arithmetic sequence must be used since there is a common difference, not a common ratio.​​ 

a-ii- Now, since it is an arithmetic sequence, let us use the nth formula of it where,​​ a=5​​ &​​ d=0.25

Un=a+n-1d

Un=5+n-10.25

Un=5+0.25n+4.75

Un=0.25n +4.75

b- So, the total amount Adina has to save for buying a computer is £350. And since her saving pattern appears to be arithmetic series, let us find the number of weeks it would take her to save £350, where​​ Sn=350,​​ a=5, and​​ d= 0.25.

Sn=n2 2a+n-1d

350=n2 25+n-10.25

700=n10+0.75n -0.25

700=n9.75+0.25

9.75+0.25 n2=700

0.25n2+9.75-700=0

n2+39n-2800=0

n=- 39±392-41 -28.02

n=36.8937…=37

n= -75.893…. (Rejected)

Since, week number should be a positive integer, accepting 37 weeks. ​​ 

Hence,​​ it will take​​ 37 weeks for Adina to save enough money to buy the computer.