**The Name ’Arithmetic’**

If *a*, *b*, and *c* are any consecutive terms of an arithmetic sequence then

*b*–

*a*=

*c*–

*b*{equating common differences}.

∴2

*b*=

*a*+

*c*

∴

*b*=½(

*a*+

*c*).

So, the middle term is the

**arithmetic mean**of the terms on either side of it.

**💎 Arithmetic Mean (A.M)**:

If *a*, *A*, *b* are three consecutive terms in an Arithmetic sequence, Then *A* is called the Arithmetic Mean (A.M) of *a* and *b*,

i.e. if *a*, *A*, *b* are in A.P. then

*A*–

*a*=

*b*–

*A*

*A*+

*A*=

*a*+

*b*

2

*A*=

*a*+

*b*

*A*=½(

*a*+

*b*).

The arithmetic mean of two numbers is equal to one half the sum of the two numbers.

If we consider three successive terms in an arithmetic sequence, namely *x*, *y*, and *z*, then since *y*–*x*= the common difference, *d*, and *z*–*y*=*d*, it follows that:

*y*–

*x*=

*z*–

*y*→

*y*=½(

*x*+

*z*).

The middle term of any three consecutive terms in an arithmetic sequence is called an arithmetic mean and is the average of the outer two terms.

That is,y=½(x+z) for any 3 consecutive termsx,y,zof an arithmetic sequence.

📌 Example 1. Find the A.M. between √5-4 and √5+4

✍ Solution:

Here *a*=√5-4; *b*=√5+4.

*A*=½(

*a*+

*b*)

*A*=½(√5-4+√5+4)

*A*=½×2√5=√5

If we need to find three consecutive temis of an arithmetic sequence, we let the numbers be

a–d,a,a+d.

If we need to find five consecutive terms of an arithmetic sequence, we let the numbers be:

a-2d,a–d,a,a+d,a+2d.

Keep ‘a’ in the middle of the sequence.

📌 Ex2. Find *k*: given that 3*k*+1, *k*, and -3 are consecutive terms of an arithmetic sequence.

✍ Solution:

Since the terms are consecutive, *k*-(3*k*+1)=-3-*k* {equating differences},

*k*-3

*k*-1=-3-

*k*

-2

*k*-1=-3-

*k*

-1+3=-

*k*+2

*k*

*k*=2

or Since the middle term is the arithmetic mean of the terms on either side of it,

*k*=½[(3

*k*+1)+(-3)]

2

*k*=3

*k*-2

*k*=2.

📌 Ex3. Three numbers are in arithmetic sequence. Their sum is 24 and their product is 494. Find the three numbers.

✍ Solution:

Let the three terms be (*a*–*d*), *a*, (*a*+*d*), which are in arithmetic sequence.

Given: Sum of the three terms =24,

*a*–

*d*)+

*a*+(

*a*+

*d*)=24

*a*–

*d*+

*a*+

*a*+

*d*=24

3

*a*=2

*a*=8.

Given: Product of the three terms =494

Thus, the three terms are 13/2, 8, 19/2.

Let’s read post •Geometric Mean Examples👈.