Question:
Find the average of even numbers from 12 to 838
Correct Answer
425
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 838
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 838 are
12, 14, 16, . . . . 838
After observing the above list of the even numbers from 12 to 838 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 838 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 838
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 838
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 838
= 12 + 838/2
= 850/2 = 425
Thus, the average of the even numbers from 12 to 838 = 425 Answer
Method (2) to find the average of the even numbers from 12 to 838
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 838 are
12, 14, 16, . . . . 838
The even numbers from 12 to 838 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 838
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 838
838 = 12 + (n – 1) × 2
⇒ 838 = 12 + 2 n – 2
⇒ 838 = 12 – 2 + 2 n
⇒ 838 = 10 + 2 n
After transposing 10 to LHS
⇒ 838 – 10 = 2 n
⇒ 828 = 2 n
After rearranging the above expression
⇒ 2 n = 828
After transposing 2 to RHS
⇒ n = 828/2
⇒ n = 414
Thus, the number of terms of even numbers from 12 to 838 = 414
This means 838 is the 414th term.
Finding the sum of the given even numbers from 12 to 838
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 838
= 414/2 (12 + 838)
= 414/2 × 850
= 414 × 850/2
= 351900/2 = 175950
Thus, the sum of all terms of the given even numbers from 12 to 838 = 175950
And, the total number of terms = 414
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 838
= 175950/414 = 425
Thus, the average of the given even numbers from 12 to 838 = 425 Answer
Similar Questions
(1) What will be the average of the first 4835 odd numbers?
(2) Find the average of odd numbers from 13 to 815
(3) Find the average of the first 2985 odd numbers.
(4) Find the average of even numbers from 4 to 974
(5) What will be the average of the first 4378 odd numbers?
(6) What is the average of the first 323 even numbers?
(7) Find the average of odd numbers from 13 to 415
(8) What is the average of the first 887 even numbers?
(9) Find the average of odd numbers from 5 to 891
(10) Find the average of odd numbers from 13 to 795