Question:
Find the average of even numbers from 8 to 624
Correct Answer
316
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 624
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 624 are
8, 10, 12, . . . . 624
After observing the above list of the even numbers from 8 to 624 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 624 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 624
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 624
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 624
= 8 + 624/2
= 632/2 = 316
Thus, the average of the even numbers from 8 to 624 = 316 Answer
Method (2) to find the average of the even numbers from 8 to 624
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 624 are
8, 10, 12, . . . . 624
The even numbers from 8 to 624 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 624
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 624
624 = 8 + (n – 1) × 2
⇒ 624 = 8 + 2 n – 2
⇒ 624 = 8 – 2 + 2 n
⇒ 624 = 6 + 2 n
After transposing 6 to LHS
⇒ 624 – 6 = 2 n
⇒ 618 = 2 n
After rearranging the above expression
⇒ 2 n = 618
After transposing 2 to RHS
⇒ n = 618/2
⇒ n = 309
Thus, the number of terms of even numbers from 8 to 624 = 309
This means 624 is the 309th term.
Finding the sum of the given even numbers from 8 to 624
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 624
= 309/2 (8 + 624)
= 309/2 × 632
= 309 × 632/2
= 195288/2 = 97644
Thus, the sum of all terms of the given even numbers from 8 to 624 = 97644
And, the total number of terms = 309
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 624
= 97644/309 = 316
Thus, the average of the given even numbers from 8 to 624 = 316 Answer
Similar Questions
(1) Find the average of the first 2418 odd numbers.
(2) Find the average of the first 4171 even numbers.
(3) Find the average of the first 3498 odd numbers.
(4) Find the average of the first 2798 odd numbers.
(5) Find the average of odd numbers from 13 to 465
(6) Find the average of the first 2517 even numbers.
(7) Find the average of even numbers from 8 to 806
(8) Find the average of even numbers from 4 to 450
(9) What is the average of the first 675 even numbers?
(10) Find the average of odd numbers from 3 to 435