Question:
Find the average of even numbers from 12 to 1066
Correct Answer
539
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1066
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1066 are
12, 14, 16, . . . . 1066
After observing the above list of the even numbers from 12 to 1066 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1066 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1066
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1066
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 12 to 1066
= 12 + 1066/2
= 1078/2 = 539
Thus, the average of the even numbers from 12 to 1066 = 539 Answer
Method (2) to find the average of the even numbers from 12 to 1066
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1066 are
12, 14, 16, . . . . 1066
The even numbers from 12 to 1066 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1066
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 12 to 1066
1066 = 12 + (n – 1) × 2
⇒ 1066 = 12 + 2 n – 2
⇒ 1066 = 12 – 2 + 2 n
⇒ 1066 = 10 + 2 n
After transposing 10 to LHS
⇒ 1066 – 10 = 2 n
⇒ 1056 = 2 n
After rearranging the above expression
⇒ 2 n = 1056
After transposing 2 to RHS
⇒ n = 1056/2
⇒ n = 528
Thus, the number of terms of even numbers from 12 to 1066 = 528
This means 1066 is the 528th term.
Finding the sum of the given even numbers from 12 to 1066
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 12 to 1066
= 528/2 (12 + 1066)
= 528/2 × 1078
= 528 × 1078/2
= 569184/2 = 284592
Thus, the sum of all terms of the given even numbers from 12 to 1066 = 284592
And, the total number of terms = 528
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 12 to 1066
= 284592/528 = 539
Thus, the average of the given even numbers from 12 to 1066 = 539 Answer
Similar Questions
(1) Find the average of even numbers from 4 to 1118
(2) Find the average of the first 1050 odd numbers.
(3) Find the average of odd numbers from 11 to 685
(4) Find the average of the first 4086 even numbers.
(5) Find the average of odd numbers from 3 to 791
(6) Find the average of odd numbers from 3 to 987
(7) Find the average of the first 3831 even numbers.
(8) Find the average of the first 3885 odd numbers.
(9) Find the average of even numbers from 10 to 1300
(10) Find the average of even numbers from 8 to 1002