Question:
Find the average of even numbers from 8 to 918
Correct Answer
463
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 918
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 918 are
8, 10, 12, . . . . 918
After observing the above list of the even numbers from 8 to 918 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 918 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 918
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 918
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 8 to 918
= 8 + 918/2
= 926/2 = 463
Thus, the average of the even numbers from 8 to 918 = 463 Answer
Method (2) to find the average of the even numbers from 8 to 918
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 918 are
8, 10, 12, . . . . 918
The even numbers from 8 to 918 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 918
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 8 to 918
918 = 8 + (n – 1) × 2
⇒ 918 = 8 + 2 n – 2
⇒ 918 = 8 – 2 + 2 n
⇒ 918 = 6 + 2 n
After transposing 6 to LHS
⇒ 918 – 6 = 2 n
⇒ 912 = 2 n
After rearranging the above expression
⇒ 2 n = 912
After transposing 2 to RHS
⇒ n = 912/2
⇒ n = 456
Thus, the number of terms of even numbers from 8 to 918 = 456
This means 918 is the 456th term.
Finding the sum of the given even numbers from 8 to 918
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 8 to 918
= 456/2 (8 + 918)
= 456/2 × 926
= 456 × 926/2
= 422256/2 = 211128
Thus, the sum of all terms of the given even numbers from 8 to 918 = 211128
And, the total number of terms = 456
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 8 to 918
= 211128/456 = 463
Thus, the average of the given even numbers from 8 to 918 = 463 Answer
Similar Questions
(1) Find the average of the first 2998 even numbers.
(2) Find the average of even numbers from 8 to 1284
(3) Find the average of the first 634 odd numbers.
(4) What is the average of the first 169 odd numbers?
(5) What is the average of the first 676 even numbers?
(6) Find the average of the first 1724 odd numbers.
(7) Find the average of the first 2003 even numbers.
(8) Find the average of the first 2363 odd numbers.
(9) What is the average of the first 1867 even numbers?
(10) Find the average of odd numbers from 13 to 921