Sequences & Series

Edexcel IAL WMA12/01/P2/Jan 2021/Q10 (Series & Sequences and their Applications)

In this question you must show detailed reasoning.​​ 

Owen wants to train for 12 weeks in preparation for running a marathon.​​ 

During the 12-week period he will run every Sunday and every Wednesday.​​ 

• ​​ On Sunday in week 1 he will run 15km​​ 

• On Sunday in week 12 he will run 37km​​ 

He considers two different 12-week training plans.​​ 

In training plan A, he will increase the distance he runs each Sunday by the same amount.​​ 

• Calculate the distance he will run on Sunday in week 5 under training plan A.​​ 

(3)​​ 

In training plan B, he will increase the distance he runs each Sunday by the same percentage.​​ 

• Calculate the distance he will run on Sunday in week 5 under training plan B.​​ 

Give your answer in km to one decimal place.​​ 

(3)​​ 

Owen will also run a fixed distance, x km, each Wednesday over the 12-week period. Given that​​ 

• x is an integer​​ 

• the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360km​​ 

• ​​ 

• ​​ find the maximum value of x, if he uses training plan A,​​ 

• find the maximum value of x, if he uses training plan B.​​ 

(5)

SOLUTION​​ 

a-​​ 

It is given that In training plan A, he will increase the distance he runs each Sunday by the same amount, which shows that there is a common difference between each successive distance travelled by the Owen. Therefore, plan A gives the arithmetic sequence.​​ 

Using the formula of Arithmetic Sequence to find the common difference, d.​​ 

Un=a+n-1d

U12=15+12-1d

37=15+12-1d

37-15=11d

22=11d

d=2

Now, finding the distance Owen runs on 5th​​ week under training plan A.​​ 

U5=15+4 x 2

U5=23

b-​​ It is given that In training plan B, he will increase the distance he runs each Sunday by the same percentage, which shows that there will be a common ratio between​​ each successive distance run by the Owen. Therefore, plan B gives the geometric ​​ sequence.​​ 

Using the formula of Geometric Sequence to find the common ratio, r.​​ 

Un=arn-1

U12=15×r12-1

37=15 r12-1

37=15r11

3715=r11

r=371511=1.08554…

Now, finding the distance Owen runs on 5th​​ week under training plan B.​​ 

U5=15 x 3715114

=20 .829

U5=20.8 km

c- i- Let us first calculate the distance Owen will travel on Sundays under plan A over 12 weeks.​​ 

Using the second formula of arithmetic series.​​ 

Sn=n2a+1

S!2=122 15+37

S!2=6 x 52

S!2=312 km

Now, this is the distance travelled only on Sundays over 12 weeks. To calculate the distance travelled on each Wednesday can be found by using the given data that the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360km. Meainng thereby,​​ 

total distance travelled over 12 weeks on Sundays and Wednesdays=total distance travelled on Sundays+total distance travelled on Wednesdays

360=312 +12x

x=4812

x=4km  

c- ii- Let us first calculate the distance, Owen will travel on Sundays under plan B over 12 weeks.​​ 

Using the second formula of geometric series, since​​ r=1.08>1.​​ 

Sn=arn-1r-1

S12=15 1.08554…12-1371511-1

S12=294.185

Now, this is the distance travelled only on Sundays over 12 weeks. To calculate the distance travelled on each Wednesday can be found by using the given data that the total distance that Owen will run on Sundays and Wednesdays over the 12 weeks will not exceed 360km. Meainng thereby,​​ 

total distance travelled over 12 weeks on Sundays and Wednesdays=total distance travelled on Sundays+total distance travelled on Wednesdays

360=294.185+12x

360-294.185=12x

x=65.81512=5.484≈5 km

x=5 km