# [Solved] If K+1 2k-1 3k+1 Are Three Consecutive Terms Of Geometric Progression, Find The Possible Values Of The Common Ratio

Given the three terms of the GP: �+1,2�−1,3�+1*K*+1,2*K*−1,3*K*+1

**Step 1:** Find the ratio using the first two terms: �=2�−1�+1*r*=*K*+12*K*−1

**Step 2:** Find the ratio using the last two terms: �=3�+12�−1*r*=2*K*−13*K*+1

**Step 3:** Equate the two expressions for �*r*: 2�−1�+1=3�+12�−1*K*+12*K*−1=2*K*−13*K*+1

Now, let’s solve for �*K* in this equation.

Finished working

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The simplified solutions for �*K* are �=0*K*=0 and �=8*K*=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*:

�=2�−1�+1*r*=*K*+12*K*−1

Let’s calculate �*r* for both values of �*K*.

Finished working

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The simplified values for the common ratio �*r* based on the values of �*K* are:

- �=−1
*r*=−1 when �=0*K*=0 - �=53
*r*=35 (or 1.67) when �=8*K*=8

Thus, the possible values for the common ratio are �=−1*r*=−1 and �=53*r*=35.