Question:
Find the average of even numbers from 4 to 242
Correct Answer
123
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 242
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 242 are
4, 6, 8, . . . . 242
After observing the above list of the even numbers from 4 to 242 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 242 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 242
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 242
The average of the numbers forming an Arithmetic Series
= The first term (a) + The last term (ℓ)/2
⇒ The average of numbers forming an Arithmetic Series = a + ℓ/2
Thus, the average of the even numbers from 4 to 242
= 4 + 242/2
= 246/2 = 123
Thus, the average of the even numbers from 4 to 242 = 123 Answer
Method (2) to find the average of the even numbers from 4 to 242
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 242 are
4, 6, 8, . . . . 242
The even numbers from 4 to 242 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 242
The Average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, to find the average of the given numbers, first, we need to find their sum and the total number of given numbers
Finding the number of terms
For an Arithmetic Series, the nth term
an = a + (n – 1) d
Where
a = First term
d = Common difference
n = number of terms
an = nth term
Thus, for the given series of the even numbers from 4 to 242
242 = 4 + (n – 1) × 2
⇒ 242 = 4 + 2 n – 2
⇒ 242 = 4 – 2 + 2 n
⇒ 242 = 2 + 2 n
After transposing 2 to LHS
⇒ 242 – 2 = 2 n
⇒ 240 = 2 n
After rearranging the above expression
⇒ 2 n = 240
After transposing 2 to RHS
⇒ n = 240/2
⇒ n = 120
Thus, the number of terms of even numbers from 4 to 242 = 120
This means 242 is the 120th term.
Finding the sum of the given even numbers from 4 to 242
The sum of all terms (S) in an Arithmetic Series
= n/2 (a + ℓ)
Where, n = number of terms
a = First term
And, ℓ = Last term
Thus, the sum of all terms (S) of the given even numbers from 4 to 242
= 120/2 (4 + 242)
= 120/2 × 246
= 120 × 246/2
= 29520/2 = 14760
Thus, the sum of all terms of the given even numbers from 4 to 242 = 14760
And, the total number of terms = 120
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given even numbers from 4 to 242
= 14760/120 = 123
Thus, the average of the given even numbers from 4 to 242 = 123 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 1327
(2) Find the average of odd numbers from 11 to 235
(3) Find the average of even numbers from 12 to 230
(4) What is the average of the first 1140 even numbers?
(5) Find the average of the first 2896 odd numbers.
(6) Find the average of odd numbers from 15 to 949
(7) Find the average of odd numbers from 13 to 699
(8) What will be the average of the first 4595 odd numbers?
(9) What is the average of the first 894 even numbers?
(10) Find the average of the first 600 odd numbers.