Question:
Find the average of even numbers from 6 to 1468
Correct Answer
737
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1468
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1468 are
6, 8, 10, . . . . 1468
After observing the above list of the even numbers from 6 to 1468 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1468 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1468
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1468
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 1468
= 6 + 1468/2
= 1474/2 = 737
Thus, the average of the even numbers from 6 to 1468 = 737 Answer
Method (2) to find the average of the even numbers from 6 to 1468
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1468 are
6, 8, 10, . . . . 1468
The even numbers from 6 to 1468 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1468
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 1468
1468 = 6 + (n – 1) × 2
⇒ 1468 = 6 + 2 n – 2
⇒ 1468 = 6 – 2 + 2 n
⇒ 1468 = 4 + 2 n
After transposing 4 to LHS
⇒ 1468 – 4 = 2 n
⇒ 1464 = 2 n
After rearranging the above expression
⇒ 2 n = 1464
After transposing 2 to RHS
⇒ n = 1464/2
⇒ n = 732
Thus, the number of terms of even numbers from 6 to 1468 = 732
This means 1468 is the 732th term.
Finding the sum of the given even numbers from 6 to 1468
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 1468
= 732/2 (6 + 1468)
= 732/2 × 1474
= 732 × 1474/2
= 1078968/2 = 539484
Thus, the sum of all terms of the given even numbers from 6 to 1468 = 539484
And, the total number of terms = 732
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 1468
= 539484/732 = 737
Thus, the average of the given even numbers from 6 to 1468 = 737 Answer
Similar Questions
(1) Find the average of even numbers from 4 to 1018
(2) What is the average of the first 95 odd numbers?
(3) Find the average of the first 2899 odd numbers.
(4) Find the average of the first 2908 odd numbers.
(5) Find the average of even numbers from 10 to 1466
(6) Find the average of odd numbers from 5 to 49
(7) Find the average of even numbers from 4 to 244
(8) What is the average of the first 1373 even numbers?
(9) Find the average of odd numbers from 13 to 1099
(10) Find the average of odd numbers from 9 to 1211