Question:
Find the average of even numbers from 8 to 1230
Correct Answer
619
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1230
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1230 are
8, 10, 12, . . . . 1230
After observing the above list of the even numbers from 8 to 1230 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1230 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1230
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1230
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 1230
= 8 + 1230/2
= 1238/2 = 619
Thus, the average of the even numbers from 8 to 1230 = 619 Answer
Method (2) to find the average of the even numbers from 8 to 1230
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1230 are
8, 10, 12, . . . . 1230
The even numbers from 8 to 1230 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1230
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 1230
1230 = 8 + (n – 1) × 2
⇒ 1230 = 8 + 2 n – 2
⇒ 1230 = 8 – 2 + 2 n
⇒ 1230 = 6 + 2 n
After transposing 6 to LHS
⇒ 1230 – 6 = 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 8 to 1230 = 612
This means 1230 is the 612th term.
Finding the sum of the given even numbers from 8 to 1230
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 1230
= 612/2 (8 + 1230)
= 612/2 × 1238
= 612 × 1238/2
= 757656/2 = 378828
Thus, the sum of all terms of the given even numbers from 8 to 1230 = 378828
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 8 to 1230
= 378828/612 = 619
Thus, the average of the given even numbers from 8 to 1230 = 619 Answer
Similar Questions
(1) Find the average of the first 612 odd numbers.
(2) Find the average of the first 2024 even numbers.
(3) Find the average of odd numbers from 13 to 1153
(4) Find the average of even numbers from 6 to 1310
(5) Find the average of the first 254 odd numbers.
(6) Find the average of the first 3334 odd numbers.
(7) Find the average of the first 2105 odd numbers.
(8) Find the average of the first 2318 even numbers.
(9) Find the average of even numbers from 12 to 804
(10) Find the average of odd numbers from 3 to 349