Sequences & Series

Edexcel IAL WMA12/01/P2/Jan 2022/Q8 (Arithmetic & Geometric Series)

A metal post is repeatedly hit in order to drive it into the ground.

Given that​​ 

• on the 1st hit, the post is driven 100mm into the ground​​ 

• on the 2nd hit, the post is driven an additional 98mm into the ground​​ 

• on the 3rd hit, the post is driven an additional 96mm into the ground​​ 

• the additional distances the post travels on each subsequent hit form an arithmetic sequence​​ 

• show that the post is driven an additional 62mm into the ground with the 20th hit.​​ 

(1)​​ 

• Find the total distance that the post has been driven into the ground after 20 hits.​​ 

(2)​​ 

Given that for each subsequent hit after the 20th hit​​ 

• ​​ the additional distances the post travels form a geometric sequence with common ratio r​​ 

• on the 22nd hit, the post is driven an additional 60mm into the ground​​ 

• find the value of r, giving your answer to 3 decimal places.​​ 

(2)

After a total of N hits, the post will have been driven more than 3m into the ground.​​ 

• Find, showing all steps in your working, the smallest possible value of N.​​ 

(4)

SOLUTION​​ 

a- The sequence appears to be​​ 

100, 98, 96, …

Since the question says that the additional distances form an arithmetic sequence, let us use the formula of nth temr​​ of the guven progression to show that the additional distance driven by the post on the 20th​​ hit is 62mm.​​ 

Un=a+n-1d

Here,​​ n=20;​​ a=100; and   d=-2

 U20=100+19-2=100-38

U20=62 mm

b- To find the total distance the post has driven after 20 hits is found by applying the​​ Sn formula of arithmetiuc progression.​​ 

Using the 2nd​​ formula since we know both the terms, the first and the last term of the arithmetic progression.​​ 

Sn=n2a+l

Where​​ a=100, l=62 & n=20.​​ 

S20=202 100+62=10162

S20=1620mm  

c- Now, after 20 hits, it forms geometric progression. Remember, the first term of the geometric term will be 62.​​ 

The expression for common ration, r.

r=U22U21=U21U20

60x=x62

60 x  62=x2

2730=x2

3720=x2

x=2930

Now finding the value of r, where​​ 

r=U22U21=60x=602930

r=0.98373…

r=0.984

d-​​ After 20 hits,​​ the total distance the post has driven was calculated in part b as​​ 1620 mm.​​ 

Whereas it says in the question that after total N hits the total distance the post has been driven is 3m or 3000mm. Now, since after 20 hits the rogression is geometric, the sum of geometric series starting from 20th​​ hit upto n terms can be found by subtraction 1529 (sum of first 20 hoits which forms arithmetic series) from 3000.​​ 

Sn=3000-1620=1380mm

​​ Let us find the n value using the formula of​​ Sn=arn-1r-1, since​​ r=0.984<1.​​ 

Sn=arn-1r-1

1380=620.984n-10.984-1

13800.984-1=620.984n-1

13800.984-162+1=0.984n

Taking log on both sides so that n becomes the coefficient of 0.984 and n can be found.​​ 

log⁡13800.984-162+1=log⁡0.984n

-0.19=nlog⁡0.984

n=-0.19log⁡0.984

n=27.89…

n=28

These are the hits forming geometric progression.​​ 

The total number of hits N are given as​​ 

N=20 hits forming arithmetic progression+28 hits forming geometric progression

N=20+28

N=48