3) 1/9, 1/3, 1, 3, ... a. Solve for the 8th term: $a_8 = 243$ b. Solve for the 3rd term: $a_3 = 1$
Answer (1)
To solve the problem, we need to determine the pattern of the sequence: 9 1 , 3 1 , 1 , 3 , … .
This is a geometric sequence because each term is obtained by multiplying the previous term by a constant ratio. Let's determine the ratio:
Divide the second term by the first term: Common ratio = 9 1 3 1 = 3 1 × 9 = 3
Now, verify this ratio with the next terms:
3 1 × 3 = 1
1 × 3 = 3
The common ratio is 3 , confirming that this is a geometric sequence with the ratio of 3 .
The general formula for the n -th term of a geometric sequence is: a n = a 1 × r ( n − 1 )
Where:
a 1 = 9 1 (the first term)
r = 3 (the common ratio)
Let's solve the parts of the question:
a. Solve for the 8th term a 8 : a 8 = 9 1 × 3 ( 8 − 1 ) = 9 1 × 3 7 3 7 = 2187 , so a 8 = 9 2187 = 243
The 8th term is 243 .
b. Solve for the 3rd term a 3 : As given, a 3 = 1 , which was verified using the ratio, as follows: a 3 = 9 1 × 3 ( 3 − 1 ) = 9 1 × 9 = 1
The 3rd term a 3 is indeed 1 , confirming the provided information.
