Question : Find the average of even numbers from 4 to 932
Correct Answer 468
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 932
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 932 are
4, 6, 8, . . . . 932
After observing the above list of the even numbers from 4 to 932 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 932 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 932
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 932
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 4 to 932
= 4 + 932/2
= 936/2 = 468
Thus, the average of the even numbers from 4 to 932 = 468 Answer
Method (2) to find the average of the even numbers from 4 to 932
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 932 are
4, 6, 8, . . . . 932
The even numbers from 4 to 932 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 932
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 4 to 932
932 = 4 + (n – 1) × 2
⇒ 932 = 4 + 2 n – 2
⇒ 932 = 4 – 2 + 2 n
⇒ 932 = 2 + 2 n
After transposing 2 to LHS
⇒ 932 – 2 = 2 n
⇒ 930 = 2 n
After rearranging the above expression
⇒ 2 n = 930
After transposing 2 to RHS
⇒ n = 930/2
⇒ n = 465
Thus, the number of terms of even numbers from 4 to 932 = 465
This means 932 is the 465th term.
Finding the sum of the given even numbers from 4 to 932
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 4 to 932
= 465/2 (4 + 932)
= 465/2 × 936
= 465 × 936/2
= 435240/2 = 217620
Thus, the sum of all terms of the given even numbers from 4 to 932 = 217620
And, the total number of terms = 465
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 4 to 932
= 217620/465 = 468
Thus, the average of the given even numbers from 4 to 932 = 468 Answer
Similar Questions
(1) Find the average of the first 2988 even numbers.
(2) Find the average of even numbers from 6 to 870
(3) Find the average of even numbers from 8 to 1428
(4) Find the average of even numbers from 10 to 238
(5) Find the average of the first 3346 odd numbers.
(6) Find the average of the first 3768 odd numbers.
(7) Find the average of even numbers from 10 to 182
(8) Find the average of the first 2134 even numbers.
(9) Find the average of odd numbers from 5 to 411
(10) Find the average of the first 4846 even numbers.