Question:
Find the average of even numbers from 10 to 476
Correct Answer
243
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 476
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 476 are
10, 12, 14, . . . . 476
After observing the above list of the even numbers from 10 to 476 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 476 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 476
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 476
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 10 to 476
= 10 + 476/2
= 486/2 = 243
Thus, the average of the even numbers from 10 to 476 = 243 Answer
Method (2) to find the average of the even numbers from 10 to 476
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 476 are
10, 12, 14, . . . . 476
The even numbers from 10 to 476 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 476
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 10 to 476
476 = 10 + (n – 1) × 2
⇒ 476 = 10 + 2 n – 2
⇒ 476 = 10 – 2 + 2 n
⇒ 476 = 8 + 2 n
After transposing 8 to LHS
⇒ 476 – 8 = 2 n
⇒ 468 = 2 n
After rearranging the above expression
⇒ 2 n = 468
After transposing 2 to RHS
⇒ n = 468/2
⇒ n = 234
Thus, the number of terms of even numbers from 10 to 476 = 234
This means 476 is the 234th term.
Finding the sum of the given even numbers from 10 to 476
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 10 to 476
= 234/2 (10 + 476)
= 234/2 × 486
= 234 × 486/2
= 113724/2 = 56862
Thus, the sum of all terms of the given even numbers from 10 to 476 = 56862
And, the total number of terms = 234
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 10 to 476
= 56862/234 = 243
Thus, the average of the given even numbers from 10 to 476 = 243 Answer
Similar Questions
(1) What is the average of the first 1394 even numbers?
(2) Find the average of the first 1943 odd numbers.
(3) What is the average of the first 894 even numbers?
(4) Find the average of odd numbers from 7 to 1045
(5) What is the average of the first 687 even numbers?
(6) Find the average of the first 3732 even numbers.
(7) Find the average of the first 2554 odd numbers.
(8) Find the average of the first 1275 odd numbers.
(9) What is the average of the first 1531 even numbers?
(10) Find the average of odd numbers from 13 to 585