Sequences & Series

Edexcel IAL WMA12/01/P2/Oct 2021/Q2 (Series – Recurrence Relationships – Sigma Notation)

A sequence is defined by​​ 

u1 = 6 

un+1 = kun + 3 

where k is a positive constant.​​ 

• Find, in terms of k, an expression for​​ U3

(2)​​ 

​​ Given that​​ ∑n=13Un  

• find the value of k.​​ 

(3)

SOLUTION​​ 

a- To find the expression for​​ U3, first we will substitute the value of n as 1 in the given expression of​​ un+1​​ to get​​ U3, and at last we will place n as 2 in the formula to find the​​ U3.​​ 

Un+1=kun+3

U1+1=ku1+3

u2=ku1+3

Where​​ u1=6

u2=k6+3

Now, finding​​ U3., by subtituting the value of n as 2.

Un+1=kun+3

U2+1=ku2+3

U3=ku2+3

Substituting the value of​​ u2= 6k+3​​ in the expression of​​ U3.

U3=k6k+3+3

U3=6k2+3k+3

U3=6k2+3k+3   

b-

∑n=13un=u1+u2+u3=117

6+6k+3+6k2+3k+3=117

6k2+9k+12=117

6k2+9k-105=0

2k2+3k=35= 0

2k+7=0            k=-5

Since​​ k​​ is a positive constant as given in the question; hence,​​ k=72