Question:
Find the average of even numbers from 4 to 622
Correct Answer
313
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 622
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 622 are
4, 6, 8, . . . . 622
After observing the above list of the even numbers from 4 to 622 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 622 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 622
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 622
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 622
= 4 + 622/2
= 626/2 = 313
Thus, the average of the even numbers from 4 to 622 = 313 Answer
Method (2) to find the average of the even numbers from 4 to 622
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 622 are
4, 6, 8, . . . . 622
The even numbers from 4 to 622 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 622
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 622
622 = 4 + (n – 1) × 2
⇒ 622 = 4 + 2 n – 2
⇒ 622 = 4 – 2 + 2 n
⇒ 622 = 2 + 2 n
After transposing 2 to LHS
⇒ 622 – 2 = 2 n
⇒ 620 = 2 n
After rearranging the above expression
⇒ 2 n = 620
After transposing 2 to RHS
⇒ n = 620/2
⇒ n = 310
Thus, the number of terms of even numbers from 4 to 622 = 310
This means 622 is the 310th term.
Finding the sum of the given even numbers from 4 to 622
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 622
= 310/2 (4 + 622)
= 310/2 × 626
= 310 × 626/2
= 194060/2 = 97030
Thus, the sum of all terms of the given even numbers from 4 to 622 = 97030
And, the total number of terms = 310
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 622
= 97030/310 = 313
Thus, the average of the given even numbers from 4 to 622 = 313 Answer
Similar Questions
(1) Find the average of the first 4831 even numbers.
(2) Find the average of even numbers from 8 to 816
(3) Find the average of even numbers from 12 to 1346
(4) Find the average of even numbers from 12 to 938
(5) Find the average of odd numbers from 7 to 435
(6) Find the average of odd numbers from 13 to 1199
(7) Find the average of even numbers from 12 to 996
(8) Find the average of odd numbers from 7 to 1049
(9) What is the average of the first 554 even numbers?
(10) Find the average of even numbers from 4 to 1388