The seventh term, u7, of a geometric sequence is 108. The eighth term, u8, of the sequence is 36.

Question

 The seventh term, u7, of a geometric sequence is 108. The eighth term, u8, of the sequence is 36.

 

(a) Write down the common ratio of the sequence. 

 

(b) Find u1.

 

The sum of the first k terms in the sequence is 118 096.

(c) Find the value of k.

Answer 1

(a) r = u₈ / u₇ = 36/108 = 1/3

 

(b) u₁ = u₇ / r⁶ = 3⁶ · 108 = 78732

 

(c) Start with u_n =  78732(1/3)^n-1

 

∑^k_i=1 u_i = 118096

 

78732 [1 - (1/3)^k] / [1 - 1/3] = 118096

 

118098 [1 - (1/3)^k] = 118096

 

1 - (1/3)^k = 118096 / 118098 = 59048 / 59049

 

(1/3)^k = 1/59049

 

3^k = 59049 = 3¹⁰

 

k = 10

 

An easier way to tell the answer in this case is to see that u₁₀ = 4 is not a multiple of 3 and the terms before are and terms after are not integers.

 

Since 118096 is an integer but not a multiple of 3 because 1+1+8+0+9+6 = 25 is not a multiple of 3, we must have k=10.