Question:
Find the average of even numbers from 6 to 1300
Correct Answer
653
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1300
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1300 are
6, 8, 10, . . . . 1300
After observing the above list of the even numbers from 6 to 1300 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1300 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1300
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1300
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 1300
= 6 + 1300/2
= 1306/2 = 653
Thus, the average of the even numbers from 6 to 1300 = 653 Answer
Method (2) to find the average of the even numbers from 6 to 1300
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1300 are
6, 8, 10, . . . . 1300
The even numbers from 6 to 1300 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1300
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 1300
1300 = 6 + (n – 1) × 2
⇒ 1300 = 6 + 2 n – 2
⇒ 1300 = 6 – 2 + 2 n
⇒ 1300 = 4 + 2 n
After transposing 4 to LHS
⇒ 1300 – 4 = 2 n
⇒ 1296 = 2 n
After rearranging the above expression
⇒ 2 n = 1296
After transposing 2 to RHS
⇒ n = 1296/2
⇒ n = 648
Thus, the number of terms of even numbers from 6 to 1300 = 648
This means 1300 is the 648th term.
Finding the sum of the given even numbers from 6 to 1300
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 1300
= 648/2 (6 + 1300)
= 648/2 × 1306
= 648 × 1306/2
= 846288/2 = 423144
Thus, the sum of all terms of the given even numbers from 6 to 1300 = 423144
And, the total number of terms = 648
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 1300
= 423144/648 = 653
Thus, the average of the given even numbers from 6 to 1300 = 653 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 539
(2) Find the average of the first 2278 even numbers.
(3) Find the average of the first 1390 odd numbers.
(4) Find the average of odd numbers from 9 to 397
(5) Find the average of even numbers from 6 to 694
(6) Find the average of the first 2400 odd numbers.
(7) Find the average of odd numbers from 9 to 1409
(8) Find the average of odd numbers from 3 to 1043
(9) Find the average of odd numbers from 3 to 243
(10) Find the average of the first 2325 odd numbers.