Question:
Find the average of even numbers from 8 to 426
Correct Answer
217
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 426
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 426 are
8, 10, 12, . . . . 426
After observing the above list of the even numbers from 8 to 426 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 426 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 426
The First Term (a) = 8
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 8 to 426
= 8 + 426/2
= 434/2 = 217
Thus, the average of the even numbers from 8 to 426 = 217 Answer
Method (2) to find the average of the even numbers from 8 to 426
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 426 are
8, 10, 12, . . . . 426
The even numbers from 8 to 426 form an Arithmetic Series in which
The First Term (a) = 8
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 8 to 426
426 = 8 + (n – 1) × 2
⇒ 426 = 8 + 2 n – 2
⇒ 426 = 8 – 2 + 2 n
⇒ 426 = 6 + 2 n
After transposing 6 to LHS
⇒ 426 – 6 = 2 n
⇒ 420 = 2 n
After rearranging the above expression
⇒ 2 n = 420
After transposing 2 to RHS
⇒ n = 420/2
⇒ n = 210
Thus, the number of terms of even numbers from 8 to 426 = 210
This means 426 is the 210th term.
Finding the sum of the given even numbers from 8 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 8 to 426
= 210/2 (8 + 426)
= 210/2 × 434
= 210 × 434/2
= 91140/2 = 45570
Thus, the sum of all terms of the given even numbers from 8 to 426 = 45570
And, the total number of terms = 210
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 426
= 45570/210 = 217
Thus, the average of the given even numbers from 8 to 426 = 217 Answer
Similar Questions
(1) Find the average of odd numbers from 3 to 1147
(2) Find the average of the first 4767 even numbers.
(3) Find the average of the first 891 odd numbers.
(4) Find the average of even numbers from 6 to 1458
(5) Find the average of odd numbers from 15 to 965
(6) Find the average of even numbers from 6 to 810
(7) Find the average of odd numbers from 11 to 945
(8) Find the average of the first 2623 odd numbers.
(9) Find the average of odd numbers from 13 to 1341
(10) Find the average of even numbers from 8 to 612