If the sum of all even numbers from 21 to 199 is added to 11 observations whose mean value is n, then the mean value of new set becomes 99. Find the value of n.

Given:

The sum of even numbers from 21 to 199 is added to 11 observations whose mean value is n.

The mean of the new set of numbers = 99.

Formula used:

(1) Sum of n numbers in A.P.

S = 

Where, 

a, is the value of the first term

l, is the value of the last term

n, is the number of terms

S, is the sum of n numbers in A.P

(2) The value of the last term in A.P.

l = a + (n - 1)d

Where, 

a, is the value of the first term

d, is the common difference between two terms

n, is the number of terms

l, is the value of the last term

Calculation:

Let n be the number of even terms between 21 to 199.

The value of the first even number (between 21 to 199), a = 22

The value of the last even number (between 21 to 199), l = 198

The value of the common difference between two even numbers, d = 2

Now,

⇒ 198 = 22 + (n - 1) × 2

⇒ 198 = 22 + (n - 1)2

⇒ 176 = (n - 1)2

⇒ (n - 1) = 88

⇒ n = 89

Now,

Let S be the sum of all even numbers between 21 to 199.

⇒ S = 

⇒ S = 9790

Now, 

The average of 11 observations = n

The sum of all 11 observations = 11n

According to the question,

⇒  = 99

⇒  = 99

⇒ 9790 + 11n = 9900

⇒ 11n = 110

⇒ n = 10

∴ The required answer is 10.

Additional InformationFormula is used to find the average of numbers when the first and last term is known.

A = 

Where, 

a, is the first term of the Arithmetic Progression

l, is the last term of the Arithmetic Progression

A, is the average of Arithmetic Progression from a to l.

Note: The above formula is only applied for Arithmetic Progression.

If the successive terms have a common difference as a non-zero constant, then that sequence can be termed an Arithmetic sequence.