Question:
Find the average of even numbers from 6 to 1436
Correct Answer
721
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1436
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1436 are
6, 8, 10, . . . . 1436
After observing the above list of the even numbers from 6 to 1436 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1436 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1436
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1436
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 6 to 1436
= 6 + 1436/2
= 1442/2 = 721
Thus, the average of the even numbers from 6 to 1436 = 721 Answer
Method (2) to find the average of the even numbers from 6 to 1436
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1436 are
6, 8, 10, . . . . 1436
The even numbers from 6 to 1436 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1436
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 6 to 1436
1436 = 6 + (n – 1) × 2
⇒ 1436 = 6 + 2 n – 2
⇒ 1436 = 6 – 2 + 2 n
⇒ 1436 = 4 + 2 n
After transposing 4 to LHS
⇒ 1436 – 4 = 2 n
⇒ 1432 = 2 n
After rearranging the above expression
⇒ 2 n = 1432
After transposing 2 to RHS
⇒ n = 1432/2
⇒ n = 716
Thus, the number of terms of even numbers from 6 to 1436 = 716
This means 1436 is the 716th term.
Finding the sum of the given even numbers from 6 to 1436
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 6 to 1436
= 716/2 (6 + 1436)
= 716/2 × 1442
= 716 × 1442/2
= 1032472/2 = 516236
Thus, the sum of all terms of the given even numbers from 6 to 1436 = 516236
And, the total number of terms = 716
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 6 to 1436
= 516236/716 = 721
Thus, the average of the given even numbers from 6 to 1436 = 721 Answer
Similar Questions
(1) Find the average of the first 3569 even numbers.
(2) Find the average of even numbers from 6 to 1748
(3) Find the average of odd numbers from 15 to 637
(4) Find the average of the first 627 odd numbers.
(5) Find the average of even numbers from 10 to 990
(6) Find the average of odd numbers from 7 to 851
(7) Find the average of even numbers from 4 to 874
(8) Find the average of even numbers from 10 to 138
(9) Find the average of even numbers from 8 to 1066
(10) Find the average of odd numbers from 13 to 407