Question : Find the average of even numbers from 4 to 966
Correct Answer 485
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 966
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 966 are
4, 6, 8, . . . . 966
After observing the above list of the even numbers from 4 to 966 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 966 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 966
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 966
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 966
= 4 + 966/2
= 970/2 = 485
Thus, the average of the even numbers from 4 to 966 = 485 Answer
Method (2) to find the average of the even numbers from 4 to 966
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 966 are
4, 6, 8, . . . . 966
The even numbers from 4 to 966 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 966
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 966
966 = 4 + (n – 1) × 2
⇒ 966 = 4 + 2 n – 2
⇒ 966 = 4 – 2 + 2 n
⇒ 966 = 2 + 2 n
After transposing 2 to LHS
⇒ 966 – 2 = 2 n
⇒ 964 = 2 n
After rearranging the above expression
⇒ 2 n = 964
After transposing 2 to RHS
⇒ n = 964/2
⇒ n = 482
Thus, the number of terms of even numbers from 4 to 966 = 482
This means 966 is the 482th term.
Finding the sum of the given even numbers from 4 to 966
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 966
= 482/2 (4 + 966)
= 482/2 × 970
= 482 × 970/2
= 467540/2 = 233770
Thus, the sum of all terms of the given even numbers from 4 to 966 = 233770
And, the total number of terms = 482
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 966
= 233770/482 = 485
Thus, the average of the given even numbers from 4 to 966 = 485 Answer
Similar Questions
(1) What is the average of the first 305 even numbers?
(2) What will be the average of the first 4749 odd numbers?
(3) Find the average of odd numbers from 7 to 1283
(4) Find the average of even numbers from 6 to 1096
(5) Find the average of the first 4608 even numbers.
(6) Find the average of the first 2821 odd numbers.
(7) Find the average of the first 4626 even numbers.
(8) Find the average of even numbers from 4 to 792
(9) Find the average of odd numbers from 5 to 91
(10) Find the average of odd numbers from 13 to 749