Question:
Find the average of even numbers from 6 to 1228
Correct Answer
617
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 1228
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 1228 are
6, 8, 10, . . . . 1228
After observing the above list of the even numbers from 6 to 1228 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 1228 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 1228
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1228
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 6 to 1228
= 6 + 1228/2
= 1234/2 = 617
Thus, the average of the even numbers from 6 to 1228 = 617 Answer
Method (2) to find the average of the even numbers from 6 to 1228
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 1228 are
6, 8, 10, . . . . 1228
The even numbers from 6 to 1228 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 1228
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 6 to 1228
1228 = 6 + (n – 1) × 2
⇒ 1228 = 6 + 2 n – 2
⇒ 1228 = 6 – 2 + 2 n
⇒ 1228 = 4 + 2 n
After transposing 4 to LHS
⇒ 1228 – 4 = 2 n
⇒ 1224 = 2 n
After rearranging the above expression
⇒ 2 n = 1224
After transposing 2 to RHS
⇒ n = 1224/2
⇒ n = 612
Thus, the number of terms of even numbers from 6 to 1228 = 612
This means 1228 is the 612th term.
Finding the sum of the given even numbers from 6 to 1228
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 6 to 1228
= 612/2 (6 + 1228)
= 612/2 × 1234
= 612 × 1234/2
= 755208/2 = 377604
Thus, the sum of all terms of the given even numbers from 6 to 1228 = 377604
And, the total number of terms = 612
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 6 to 1228
= 377604/612 = 617
Thus, the average of the given even numbers from 6 to 1228 = 617 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 1163
(2) Find the average of odd numbers from 3 to 719
(3) Find the average of even numbers from 8 to 1078
(4) Find the average of odd numbers from 13 to 1449
(5) Find the average of even numbers from 12 to 642
(6) What is the average of the first 958 even numbers?
(7) Find the average of even numbers from 8 to 474
(8) Find the average of even numbers from 8 to 40
(9) What will be the average of the first 4914 odd numbers?
(10) Find the average of the first 3889 even numbers.