Question:
Find the average of even numbers from 4 to 796
Correct Answer
400
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 796
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 796 are
4, 6, 8, . . . . 796
After observing the above list of the even numbers from 4 to 796 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 796 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 796
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 796
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 796
= 4 + 796/2
= 800/2 = 400
Thus, the average of the even numbers from 4 to 796 = 400 Answer
Method (2) to find the average of the even numbers from 4 to 796
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 796 are
4, 6, 8, . . . . 796
The even numbers from 4 to 796 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 796
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 796
796 = 4 + (n – 1) × 2
⇒ 796 = 4 + 2 n – 2
⇒ 796 = 4 – 2 + 2 n
⇒ 796 = 2 + 2 n
After transposing 2 to LHS
⇒ 796 – 2 = 2 n
⇒ 794 = 2 n
After rearranging the above expression
⇒ 2 n = 794
After transposing 2 to RHS
⇒ n = 794/2
⇒ n = 397
Thus, the number of terms of even numbers from 4 to 796 = 397
This means 796 is the 397th term.
Finding the sum of the given even numbers from 4 to 796
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 796
= 397/2 (4 + 796)
= 397/2 × 800
= 397 × 800/2
= 317600/2 = 158800
Thus, the sum of all terms of the given even numbers from 4 to 796 = 158800
And, the total number of terms = 397
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 796
= 158800/397 = 400
Thus, the average of the given even numbers from 4 to 796 = 400 Answer
Similar Questions
(1) Find the average of the first 3995 odd numbers.
(2) Find the average of the first 3446 odd numbers.
(3) Find the average of even numbers from 10 to 1346
(4) Find the average of the first 4741 even numbers.
(5) Find the average of even numbers from 6 to 762
(6) Find the average of even numbers from 12 to 1152
(7) Find the average of the first 3625 even numbers.
(8) Find the average of even numbers from 12 to 1074
(9) Find the average of even numbers from 10 to 1676
(10) Find the average of the first 2459 odd numbers.