Question:
Find the average of even numbers from 8 to 552
Correct Answer
280
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 552
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 552 are
8, 10, 12, . . . . 552
After observing the above list of the even numbers from 8 to 552 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 552 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 552
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 552
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 552
= 8 + 552/2
= 560/2 = 280
Thus, the average of the even numbers from 8 to 552 = 280 Answer
Method (2) to find the average of the even numbers from 8 to 552
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 552 are
8, 10, 12, . . . . 552
The even numbers from 8 to 552 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 552
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 552
552 = 8 + (n – 1) × 2
⇒ 552 = 8 + 2 n – 2
⇒ 552 = 8 – 2 + 2 n
⇒ 552 = 6 + 2 n
After transposing 6 to LHS
⇒ 552 – 6 = 2 n
⇒ 546 = 2 n
After rearranging the above expression
⇒ 2 n = 546
After transposing 2 to RHS
⇒ n = 546/2
⇒ n = 273
Thus, the number of terms of even numbers from 8 to 552 = 273
This means 552 is the 273th term.
Finding the sum of the given even numbers from 8 to 552
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 552
= 273/2 (8 + 552)
= 273/2 × 560
= 273 × 560/2
= 152880/2 = 76440
Thus, the sum of all terms of the given even numbers from 8 to 552 = 76440
And, the total number of terms = 273
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 552
= 76440/273 = 280
Thus, the average of the given even numbers from 8 to 552 = 280 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 758
(2) Find the average of the first 1952 odd numbers.
(3) Find the average of odd numbers from 11 to 1127
(4) Find the average of even numbers from 12 to 514
(5) Find the average of even numbers from 6 to 1652
(6) Find the average of even numbers from 10 to 1150
(7) What is the average of the first 1698 even numbers?
(8) Find the average of odd numbers from 13 to 1297
(9) What will be the average of the first 4565 odd numbers?
(10) Find the average of even numbers from 4 to 538