Sequences & Series

Edexcel IAL WMA12/01/P2/Oct 2022/Q8 (Series & Sequences, Geometric Series, Infinity)

A geometric sequence has first term a and common ratio r​​ 

Given that​​ S∞ = 3a​​ 

• show that​​ r =23​​ 

(2)

Given also that​​ 

u2 – u4 = 16 

where uk​​ is the kth​​ term of this sequence,​​ 

• find the value of S10​​ giving your answer to one decimal place.​​ 

(5)

SOLUTION

a- Using the sum to infinity of a convergent geometric series formula.​​ 

S∞=a1-r 

Where,​​ S∞=3a 

3a=a1-r

1-r=a3a

1-r=13

1-13=r

r=23

b- Since​​ Un=arn-1 or Uk=ark-1,​​ 

when​​ k=2

U2=ar2-1

u2=a 231=2a3

And, when​​ k=4

u4=ar4-1

u4=a 233=8a27

It is given that​​ 

u2-u4=16

2a3-8a27=16

18a27-8a27=16

10a27=16

a=16 x 2710=2165

a=43.2

Now, putting the value of a and r in the​​ Sn, to find the sum of series upto​​ S10. And, since​​ r=23<0, the formula of sum of first​​ n​​ geometric series is found by

Sn=a1-rn1-r

S10=43.2 1-23101-23=127.352…

S10=127.4