Question:
Find the average of even numbers from 12 to 926
Correct Answer
469
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 926
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 926 are
12, 14, 16, . . . . 926
After observing the above list of the even numbers from 12 to 926 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 926 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 926
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 926
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 926
= 12 + 926/2
= 938/2 = 469
Thus, the average of the even numbers from 12 to 926 = 469 Answer
Method (2) to find the average of the even numbers from 12 to 926
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 926 are
12, 14, 16, . . . . 926
The even numbers from 12 to 926 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 926
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 926
926 = 12 + (n – 1) × 2
⇒ 926 = 12 + 2 n – 2
⇒ 926 = 12 – 2 + 2 n
⇒ 926 = 10 + 2 n
After transposing 10 to LHS
⇒ 926 – 10 = 2 n
⇒ 916 = 2 n
After rearranging the above expression
⇒ 2 n = 916
After transposing 2 to RHS
⇒ n = 916/2
⇒ n = 458
Thus, the number of terms of even numbers from 12 to 926 = 458
This means 926 is the 458th term.
Finding the sum of the given even numbers from 12 to 926
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 926
= 458/2 (12 + 926)
= 458/2 × 938
= 458 × 938/2
= 429604/2 = 214802
Thus, the sum of all terms of the given even numbers from 12 to 926 = 214802
And, the total number of terms = 458
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 926
= 214802/458 = 469
Thus, the average of the given even numbers from 12 to 926 = 469 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 971
(2) Find the average of odd numbers from 15 to 1151
(3) Find the average of the first 2354 odd numbers.
(4) Find the average of even numbers from 12 to 1102
(5) Find the average of the first 2004 even numbers.
(6) Find the average of even numbers from 6 to 1656
(7) Find the average of the first 442 odd numbers.
(8) Find the average of the first 3714 even numbers.
(9) Find the average of the first 3184 odd numbers.
(10) Find the average of odd numbers from 5 to 741