Sequences & Series

Edexcel IAL WMA12/01/P2/June/Oct 2020/Q8 (Geometric Series Inc Proof for Sum)

A geometric series has first term a and common ratio r.​​ 

• Prove that the sum of the first n terms of this series is given by​​ 

Sn=a1-rn1-r

(3)​​ 

The second term of a geometric series is –320 and the fifth term is​​ 51225​​ ​​ 

• Find the value of the common ratio.​​ 

(2)​​ 

• Hence find the sum of the first 13 terms of the series, giving your answer to 2 decimal places.​​ 

(3)

SOLUTION​​ 

a-​​ 

Let​​ 

Sn=a+ar+ar2+ ar3+… arn-2+arn-1           eqn (1)

Multipying the equation by r.​​ 

r×Sn=r(a+ar+ar2+ ar3+… arn-1+arn)  

r×Sn=ar+ar2+ar3+ ar4+… arn-1+arn)         eqn (2)

Subtracting eqn 2 by 1, we get.​​   

Sn-rSn=a-arn

Sn1-r=a(1-rn)

Sn=a(1-rn)1-r

b- Using the given data.

U2=-320 

U5=51225

ar=-320   -1

ar4=51225    -2

Dividing eqn​​ 2 by 1.

r3=51225 -320

r3= - 8125  

r=- 8125

r=-25

c-​​ Finding the sum upto 13 terms in a geometric series, where​​ n=13, &​​ r= - 25. But ley us first find a by substituting the value of r in any eqn of the above part.

(1)=>ar=-320

a-25=-320

a=-320 x 5-2

a=800

Now, using​​ Sn​​ formula.​​ 

S13=8001--25131--25

S13=517.4324..

Answer needs to be in 2 decimal places.

S13=571.43