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Modern Maths

Progressions

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CAT 2025 Lesson : Progressions - AP Special Types

2.3 AP where n is small

In AP 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 middle term as

x

and common difference as

y

.

- For a

3

-term AP, use the terms (

x

–

y

x

, (

x

y

)

- For a

5

-term AP, use the terms (

x

–

2y

), (

x

–

y

x

, (

x

y

), (

x

2y

)

2) Where the number of terms is even, take the middle terms as x – y and x + y and common difference as

2y

.

- For a

4

-term AP, use the terms (

x

–

3y

), (

x

–

y

), (

x

y

), (

x

3y

)

- For a

6

-term AP, use the terms (

x

–

5y

), (

x

–

3y

), (

x

–

y

), (

x

y

), (

x

3y

), (

x

5y

)

Example 9

An AP has

3

terms. If the sum of terms is

24

and the product of terms is

440

. Then, what is the third term in this A.P ?

(1)

5

(2)

11

(3)

5 or 11

(4) None of the above

Solution

Let the

3

terms be (

x

–

y

x

, (

x

y

).

Sum of

3

terms =

3x

24

⇒

x = 8

Product of

3

terms =

(x - y) \times x \times(x + y) = 440

⇒

(8 - y) \times 8 \times (8 + y)=440

⇒

64-y^2 = 55

⇒

y^2 = 9

⇒

y

\pm 3

First term =

(8 + 3) \ or \ (8 – 3)

11

5

Answer:

(3) 5 \ or \ 11

Example 10

If the sum of

6

terms of an AP is

69

and the

5^{th}

term is

16

, then what is the

3^{th}

term?

Solution

Let the

6

terms be (

x

–

5y

), (

x

–

3y

), (

x

–

y

), (

x

y

), (

x

3y

), (

x

5y

)

Sum of

6

terms =

6x = 69 ⇒ x = 11.5

5^{th}

term =

x + 3y = 16

⇒

11.5 + 3y = 16

⇒

y = 1.5

\therefore

3^{th}

term =

x - y

11.5

–

1.5

10

Answer:

10

2.4 Inserting Arithmetic Means

When Arithmetic Means are inserted between two numbers, say

x

and

y

, then all these numbers together would form an Arithmetic Progression where

x

and

y

will be the first and last terms respectively.

In any AP with

n

terms, there are (

n

– 2) arithmetic means between the first and the last terms. No direct formula is required for this. Please logically apply the AP formulae where required.

Example 11

20

Arithmetic Means are inserted between

14

and

16

, then what is the sum of these Arithmetic Means?

Solution

Note that the

20

Arithmetic Means along with

14

and

16

form an AP.

Average of this AP =

\dfrac{14+16}{2} = 15

Average of the AP with just the

20

AMs will also be

15

.

Sum of the

20

AMs =

n \times Average

20 \times 15 = 300

Answer:

300

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