Back

Progressions

ALL MODULES

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}

Example 16

If the product of

4

terms of an increasing GP is

1296

, and the product of the

2^{nd}

and

4^{th}

terms is

144

. Then, what is the

1^{st}

term?

Solution

Let the

4

terms be

\dfrac{x}{y^{3}},\dfrac{x}{y},xy,xy^{3}

Product of

4

terms =

x^{4} = 1296 = 6^{4}

⇒

x = 6

Product of

2nd

and

4th

terms =

\dfrac{x}{y}\times xy^{3} = x^{2}y^{2} = 144

⇒

6^{2} \times y^{2} = 144

⇒

y = 2

\therefore

1^{st}

term =

\dfrac{x}{y^{3}} = \dfrac{6}{8} = \dfrac{3}{4}

Answer:

\dfrac{3}{4}

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.

Example 17

m

and

n

are the

2

Geometric Means inserted between

6

and

384

, then

m + n

= ?

Solution

As the numbers along with the GMs will be in GP, let the terms be

a, ar,

ar^{2}

and

ar^{3}

respectively.

1^{st}

term =

\bm{a = 6}

4^{th}

term =

ar^{3}= 384

⇒

r^{3}=\dfrac{384}{6}

⇒

r^{3}= 64

⇒

\bm{r = 4}

m + n = ar + ar^{2} = ar( 1 + r ) = 6 \times 4 \times 5

\bm{120}

Answer:

120

Loading Video....