Question:
Find the average of even numbers from 4 to 1256
Correct Answer
630
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1256
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1256 are
4, 6, 8, . . . . 1256
After observing the above list of the even numbers from 4 to 1256 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1256 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1256
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1256
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 1256
= 4 + 1256/2
= 1260/2 = 630
Thus, the average of the even numbers from 4 to 1256 = 630 Answer
Method (2) to find the average of the even numbers from 4 to 1256
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1256 are
4, 6, 8, . . . . 1256
The even numbers from 4 to 1256 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1256
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 1256
1256 = 4 + (n – 1) × 2
⇒ 1256 = 4 + 2 n – 2
⇒ 1256 = 4 – 2 + 2 n
⇒ 1256 = 2 + 2 n
After transposing 2 to LHS
⇒ 1256 – 2 = 2 n
⇒ 1254 = 2 n
After rearranging the above expression
⇒ 2 n = 1254
After transposing 2 to RHS
⇒ n = 1254/2
⇒ n = 627
Thus, the number of terms of even numbers from 4 to 1256 = 627
This means 1256 is the 627th term.
Finding the sum of the given even numbers from 4 to 1256
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 1256
= 627/2 (4 + 1256)
= 627/2 × 1260
= 627 × 1260/2
= 790020/2 = 395010
Thus, the sum of all terms of the given even numbers from 4 to 1256 = 395010
And, the total number of terms = 627
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 1256
= 395010/627 = 630
Thus, the average of the given even numbers from 4 to 1256 = 630 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 1084
(2) Find the average of the first 4616 even numbers.
(3) Find the average of even numbers from 8 to 316
(4) Find the average of odd numbers from 5 to 1381
(5) Find the average of the first 2978 odd numbers.
(6) What is the average of the first 56 odd numbers?
(7) What is the average of the first 818 even numbers?
(8) Find the average of the first 3881 odd numbers.
(9) Find the average of the first 3664 odd numbers.
(10) Find the average of odd numbers from 11 to 573