CAT 2025 Lesson : Progressions - GP Special Types

3.3 GP where n is small

In GP with a small number of terms, you can use the following format for ease in calculations.

1) Where the number of terms is odd , take the $x$ as the middle term and $y$ as the common ratio.

- For a $3$-term GP, use the terms $\dfrac{x}{y},x,xy$

- For a $5$-term GP, use the terms $\dfrac{x}{y^{2}},\dfrac{x}{y},x,xy,xy^{2}$

2) Where the number of terms is even, take $\dfrac{x}{y}$ and $xy$ as middle terms and $y^{2}$ as the common ratio.

- For a $4$-term GP, use the terms $\dfrac{x}{y^{3}},\dfrac{x}{y},xy,xy^{3}$

- For a $6$-term GP, use the terms $\dfrac{x}{y^{5}},\dfrac{x}{y^{3}},\dfrac{x}{y},xy,xy^{3},xy^{5}$

3.4 Inserting Geometric Means

When Geometric Means are inserted between two numbers, say $x$ and $y,$ then all these numbers together would form a Geometric Progression where $x$ and $y$ will be the first and last terms respectively.

In any GP with $n$ terms, there are $(n – 2)$ geometric means between the first and the last terms. No direct formula is required for this. Please logically apply the GP formulae where required.