Question : Find the average of even numbers from 12 to 628
Correct Answer 320
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 628
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 628 are
12, 14, 16, . . . . 628
After observing the above list of the even numbers from 12 to 628 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 628 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 628
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 628
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 12 to 628
= 12 + 628/2
= 640/2 = 320
Thus, the average of the even numbers from 12 to 628 = 320 Answer
Method (2) to find the average of the even numbers from 12 to 628
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 628 are
12, 14, 16, . . . . 628
The even numbers from 12 to 628 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 628
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 12 to 628
628 = 12 + (n – 1) × 2
⇒ 628 = 12 + 2 n – 2
⇒ 628 = 12 – 2 + 2 n
⇒ 628 = 10 + 2 n
After transposing 10 to LHS
⇒ 628 – 10 = 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 12 to 628 = 309
This means 628 is the 309th term.
Finding the sum of the given even numbers from 12 to 628
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 12 to 628
= 309/2 (12 + 628)
= 309/2 × 640
= 309 × 640/2
= 197760/2 = 98880
Thus, the sum of all terms of the given even numbers from 12 to 628 = 98880
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 12 to 628
= 98880/309 = 320
Thus, the average of the given even numbers from 12 to 628 = 320 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1842
(2) Find the average of the first 2037 even numbers.
(3) Find the average of the first 3498 even numbers.
(4) Find the average of even numbers from 12 to 1520
(5) Find the average of odd numbers from 13 to 1265
(6) Find the average of the first 3602 odd numbers.
(7) Find the average of even numbers from 10 to 1400
(8) Find the average of the first 4553 even numbers.
(9) Find the average of the first 2832 odd numbers.
(10) Find the average of the first 2015 even numbers.