Given the three terms of the GP: �+1,2�−1,3�+1K+1,2K−1,3K+1
Step 1: Find the ratio using the first two terms: �=2�−1�+1r=K+12K−1
Step 2: Find the ratio using the last two terms: �=3�+12�−1r=2K−13K+1
Step 3: Equate the two expressions for �r: 2�−1�+1=3�+12�−1K+12K−1=2K−13K+1
Now, let’s solve for �K in this equation.
The simplified solutions for �K are �=0K=0 and �=8K=8.
Now, using these values for �K, let’s find the corresponding values for the common ratio �r in a simplified manner.
Step 4: Plug the values of �K into the expression for �r:
Let’s calculate �r for both values of �K.
The simplified values for the common ratio �r based on the values of �K are:
- �=−1r=−1 when �=0K=0
- �=53r=35 (or 1.67) when �=8K=8
Thus, the possible values for the common ratio are �=−1r=−1 and �=53r=35.