Question

what is the arithmetic mean of the natural numbers from 10 to 30?​

Answer 1

Arithmetic mean of the natural number from 10 to 30 is 20

Given:

• Natural number from 10 to 30

To Find:

• Arithmetic mean

Solution:

• Natural Numbers are positive Integers

$\text{Mean}=\dfrac{\text{sum of all the obervations}}{\text{number of the observations}}$

Arithmetic sequence

• Sequence of terms in which difference between one term and the next is a constant.
• This is also called Arithmetic Progression AP
• Arithmetic sequence can be represented in the form :
• a, a + d  , a + 2d , …………………………, a + (n-1)d
• a = First term
• d = common difference = aₙ-aₙ₋₁
• nth term =  aₙ =  a + (n-1)d
• Sₙ = (n/2)(2a + (n - 1)d)
• Sₙ = (n/2)( a + aₙ)

Step 1:

Natural Numbers from 10 to 30 form an AP with

a = 10  , d  = 1 , last term = 30

Step 2:

Find number of term

30 = 10 + (n - 1)1

=> 20 = n - 1

=> 21 = n

Step 3:

Find Sum of  21 terms

(21/2)(10 + 30)

= 21(20)

Step 4:

Divide by number of term 21 to find arithmetic mean

21 (20)/21

= 20

Hence, Arithmetic mean of the natural number from 10 to 30 is 20