Question:
Find the average of even numbers from 12 to 254
Correct Answer
133
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 254
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 254 are
12, 14, 16, . . . . 254
After observing the above list of the even numbers from 12 to 254 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 254 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 254
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 254
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 254
= 12 + 254/2
= 266/2 = 133
Thus, the average of the even numbers from 12 to 254 = 133 Answer
Method (2) to find the average of the even numbers from 12 to 254
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 254 are
12, 14, 16, . . . . 254
The even numbers from 12 to 254 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 254
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 254
254 = 12 + (n – 1) × 2
⇒ 254 = 12 + 2 n – 2
⇒ 254 = 12 – 2 + 2 n
⇒ 254 = 10 + 2 n
After transposing 10 to LHS
⇒ 254 – 10 = 2 n
⇒ 244 = 2 n
After rearranging the above expression
⇒ 2 n = 244
After transposing 2 to RHS
⇒ n = 244/2
⇒ n = 122
Thus, the number of terms of even numbers from 12 to 254 = 122
This means 254 is the 122th term.
Finding the sum of the given even numbers from 12 to 254
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 254
= 122/2 (12 + 254)
= 122/2 × 266
= 122 × 266/2
= 32452/2 = 16226
Thus, the sum of all terms of the given even numbers from 12 to 254 = 16226
And, the total number of terms = 122
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 254
= 16226/122 = 133
Thus, the average of the given even numbers from 12 to 254 = 133 Answer
Similar Questions
(1) What will be the average of the first 4568 odd numbers?
(2) Find the average of even numbers from 10 to 94
(3) Find the average of odd numbers from 15 to 1439
(4) What will be the average of the first 4862 odd numbers?
(5) Find the average of the first 4659 even numbers.
(6) Find the average of the first 2382 even numbers.
(7) Find the average of the first 3970 even numbers.
(8) Find the average of the first 973 odd numbers.
(9) Find the average of even numbers from 6 to 274
(10) Find the average of the first 3145 even numbers.