Question : Find the average of even numbers from 12 to 426
Correct Answer 219
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 426
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 426 are
12, 14, 16, . . . . 426
After observing the above list of the even numbers from 12 to 426 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 426 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 426
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 426
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 426
= 12 + 426/2
= 438/2 = 219
Thus, the average of the even numbers from 12 to 426 = 219 Answer
Method (2) to find the average of the even numbers from 12 to 426
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 426 are
12, 14, 16, . . . . 426
The even numbers from 12 to 426 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 426
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 426
426 = 12 + (n – 1) × 2
⇒ 426 = 12 + 2 n – 2
⇒ 426 = 12 – 2 + 2 n
⇒ 426 = 10 + 2 n
After transposing 10 to LHS
⇒ 426 – 10 = 2 n
⇒ 416 = 2 n
After rearranging the above expression
⇒ 2 n = 416
After transposing 2 to RHS
⇒ n = 416/2
⇒ n = 208
Thus, the number of terms of even numbers from 12 to 426 = 208
This means 426 is the 208th term.
Finding the sum of the given even numbers from 12 to 426
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 426
= 208/2 (12 + 426)
= 208/2 × 438
= 208 × 438/2
= 91104/2 = 45552
Thus, the sum of all terms of the given even numbers from 12 to 426 = 45552
And, the total number of terms = 208
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 426
= 45552/208 = 219
Thus, the average of the given even numbers from 12 to 426 = 219 Answer
Similar Questions
(1) What is the average of the first 1889 even numbers?
(2) Find the average of odd numbers from 11 to 107
(3) Find the average of odd numbers from 9 to 1453
(4) Find the average of odd numbers from 9 to 363
(5) Find the average of odd numbers from 3 to 441
(6) Find the average of even numbers from 6 to 1676
(7) Find the average of odd numbers from 9 to 1139
(8) What will be the average of the first 4484 odd numbers?
(9) Find the average of even numbers from 6 to 1908
(10) Find the average of the first 3914 odd numbers.