Question:
Find the average of even numbers from 12 to 722
Correct Answer
367
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 722
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 722 are
12, 14, 16, . . . . 722
After observing the above list of the even numbers from 12 to 722 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 722 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 722
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 722
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 722
= 12 + 722/2
= 734/2 = 367
Thus, the average of the even numbers from 12 to 722 = 367 Answer
Method (2) to find the average of the even numbers from 12 to 722
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 722 are
12, 14, 16, . . . . 722
The even numbers from 12 to 722 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 722
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 722
722 = 12 + (n – 1) × 2
⇒ 722 = 12 + 2 n – 2
⇒ 722 = 12 – 2 + 2 n
⇒ 722 = 10 + 2 n
After transposing 10 to LHS
⇒ 722 – 10 = 2 n
⇒ 712 = 2 n
After rearranging the above expression
⇒ 2 n = 712
After transposing 2 to RHS
⇒ n = 712/2
⇒ n = 356
Thus, the number of terms of even numbers from 12 to 722 = 356
This means 722 is the 356th term.
Finding the sum of the given even numbers from 12 to 722
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 722
= 356/2 (12 + 722)
= 356/2 × 734
= 356 × 734/2
= 261304/2 = 130652
Thus, the sum of all terms of the given even numbers from 12 to 722 = 130652
And, the total number of terms = 356
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 722
= 130652/356 = 367
Thus, the average of the given even numbers from 12 to 722 = 367 Answer
Similar Questions
(1) Find the average of the first 4585 even numbers.
(2) What is the average of the first 115 odd numbers?
(3) Find the average of even numbers from 6 to 1210
(4) Find the average of even numbers from 12 to 878
(5) Find the average of the first 673 odd numbers.
(6) Find the average of even numbers from 12 to 456
(7) Find the average of the first 2600 odd numbers.
(8) What will be the average of the first 4532 odd numbers?
(9) Find the average of the first 1418 odd numbers.
(10) Find the average of even numbers from 10 to 1144