Question:
Find the average of even numbers from 4 to 1330
Correct Answer
667
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1330
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1330 are
4, 6, 8, . . . . 1330
After observing the above list of the even numbers from 4 to 1330 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1330 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1330
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1330
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 1330
= 4 + 1330/2
= 1334/2 = 667
Thus, the average of the even numbers from 4 to 1330 = 667 Answer
Method (2) to find the average of the even numbers from 4 to 1330
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1330 are
4, 6, 8, . . . . 1330
The even numbers from 4 to 1330 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1330
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 1330
1330 = 4 + (n – 1) × 2
⇒ 1330 = 4 + 2 n – 2
⇒ 1330 = 4 – 2 + 2 n
⇒ 1330 = 2 + 2 n
After transposing 2 to LHS
⇒ 1330 – 2 = 2 n
⇒ 1328 = 2 n
After rearranging the above expression
⇒ 2 n = 1328
After transposing 2 to RHS
⇒ n = 1328/2
⇒ n = 664
Thus, the number of terms of even numbers from 4 to 1330 = 664
This means 1330 is the 664th term.
Finding the sum of the given even numbers from 4 to 1330
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 1330
= 664/2 (4 + 1330)
= 664/2 × 1334
= 664 × 1334/2
= 885776/2 = 442888
Thus, the sum of all terms of the given even numbers from 4 to 1330 = 442888
And, the total number of terms = 664
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 1330
= 442888/664 = 667
Thus, the average of the given even numbers from 4 to 1330 = 667 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 631
(2) Find the average of the first 1343 odd numbers.
(3) What is the average of the first 1692 even numbers?
(4) What will be the average of the first 4198 odd numbers?
(5) Find the average of odd numbers from 15 to 1799
(6) Find the average of even numbers from 12 to 692
(7) Find the average of odd numbers from 11 to 643
(8) Find the average of odd numbers from 11 to 259
(9) Find the average of the first 4189 even numbers.
(10) Find the average of the first 1610 odd numbers.