Question : Find the average of even numbers from 4 to 402
Correct Answer 203
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 402
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 402 are
4, 6, 8, . . . . 402
After observing the above list of the even numbers from 4 to 402 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 402 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 402
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 402
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 402
= 4 + 402/2
= 406/2 = 203
Thus, the average of the even numbers from 4 to 402 = 203 Answer
Method (2) to find the average of the even numbers from 4 to 402
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 402 are
4, 6, 8, . . . . 402
The even numbers from 4 to 402 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 402
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 402
402 = 4 + (n – 1) × 2
⇒ 402 = 4 + 2 n – 2
⇒ 402 = 4 – 2 + 2 n
⇒ 402 = 2 + 2 n
After transposing 2 to LHS
⇒ 402 – 2 = 2 n
⇒ 400 = 2 n
After rearranging the above expression
⇒ 2 n = 400
After transposing 2 to RHS
⇒ n = 400/2
⇒ n = 200
Thus, the number of terms of even numbers from 4 to 402 = 200
This means 402 is the 200th term.
Finding the sum of the given even numbers from 4 to 402
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 402
= 200/2 (4 + 402)
= 200/2 × 406
= 200 × 406/2
= 81200/2 = 40600
Thus, the sum of all terms of the given even numbers from 4 to 402 = 40600
And, the total number of terms = 200
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 402
= 40600/200 = 203
Thus, the average of the given even numbers from 4 to 402 = 203 Answer
Similar Questions
(1) What will be the average of the first 4860 odd numbers?
(2) Find the average of odd numbers from 5 to 679
(3) Find the average of even numbers from 6 to 1164
(4) Find the average of the first 2111 odd numbers.
(5) Find the average of odd numbers from 5 to 1173
(6) What will be the average of the first 4106 odd numbers?
(7) Find the average of even numbers from 12 to 1512
(8) Find the average of even numbers from 10 to 948
(9) Find the average of odd numbers from 7 to 937
(10) Find the average of odd numbers from 13 to 1249