Sequences

Concept: Solve the equation by simplifying the rational expression.

Step 1: Set equation.


Step 2: Cross multiply.


Step 3: Simplify.


Step 4: Factorize.


Step 5: Find valid solution.
Since , valid solution is:

Step 1:Understand the type of sequence.
The given sequence is:
This is a geometric sequence where each term is obtained by multiplying the previous term by a constant ratio.

Step 2:Identify the first term and the common ratio.
- The first term .
- The common ratio .

Step 3:Formula for the -th term of a geometric sequence.
The general formula for the -th term of a geometric sequence is:
where:
- is the first term,
- is the common ratio,
- is the term number.

Step 4:Substitute the values to find the 25th term.
We are asked to find the 25th term ( ) of the sequence:
Simplifying:

Step 5:Final Answer.
Therefore, the 25th term of the sequence is .

Step 1: Find .
We are given . To find , substitute into :
Step 2: Find .
We are given . To find , substitute into : Expanding the terms: Simplifying:
Step 3: Find .
Now, we subtract from : Simplifying: