Question : Find the average of even numbers from 8 to 1232
Correct Answer 620
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1232
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1232 are
8, 10, 12, . . . . 1232
After observing the above list of the even numbers from 8 to 1232 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1232 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1232
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1232
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 1232
= 8 + 1232/2
= 1240/2 = 620
Thus, the average of the even numbers from 8 to 1232 = 620 Answer
Method (2) to find the average of the even numbers from 8 to 1232
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1232 are
8, 10, 12, . . . . 1232
The even numbers from 8 to 1232 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1232
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 1232
1232 = 8 + (n – 1) × 2
⇒ 1232 = 8 + 2 n – 2
⇒ 1232 = 8 – 2 + 2 n
⇒ 1232 = 6 + 2 n
After transposing 6 to LHS
⇒ 1232 – 6 = 2 n
⇒ 1226 = 2 n
After rearranging the above expression
⇒ 2 n = 1226
After transposing 2 to RHS
⇒ n = 1226/2
⇒ n = 613
Thus, the number of terms of even numbers from 8 to 1232 = 613
This means 1232 is the 613th term.
Finding the sum of the given even numbers from 8 to 1232
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 1232
= 613/2 (8 + 1232)
= 613/2 × 1240
= 613 × 1240/2
= 760120/2 = 380060
Thus, the sum of all terms of the given even numbers from 8 to 1232 = 380060
And, the total number of terms = 613
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 1232
= 380060/613 = 620
Thus, the average of the given even numbers from 8 to 1232 = 620 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 74
(2) Find the average of even numbers from 10 to 1378
(3) What is the average of the first 742 even numbers?
(4) Find the average of even numbers from 12 to 1122
(5) Find the average of odd numbers from 15 to 1079
(6) Find the average of even numbers from 10 to 1846
(7) Find the average of the first 3308 even numbers.
(8) What will be the average of the first 4002 odd numbers?
(10) Find the average of the first 3368 odd numbers.