Question:
Find the average of even numbers from 8 to 1132
Correct Answer
570
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1132
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1132 are
8, 10, 12, . . . . 1132
After observing the above list of the even numbers from 8 to 1132 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1132 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1132
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1132
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 1132
= 8 + 1132/2
= 1140/2 = 570
Thus, the average of the even numbers from 8 to 1132 = 570 Answer
Method (2) to find the average of the even numbers from 8 to 1132
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1132 are
8, 10, 12, . . . . 1132
The even numbers from 8 to 1132 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1132
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 1132
1132 = 8 + (n – 1) × 2
⇒ 1132 = 8 + 2 n – 2
⇒ 1132 = 8 – 2 + 2 n
⇒ 1132 = 6 + 2 n
After transposing 6 to LHS
⇒ 1132 – 6 = 2 n
⇒ 1126 = 2 n
After rearranging the above expression
⇒ 2 n = 1126
After transposing 2 to RHS
⇒ n = 1126/2
⇒ n = 563
Thus, the number of terms of even numbers from 8 to 1132 = 563
This means 1132 is the 563th term.
Finding the sum of the given even numbers from 8 to 1132
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 1132
= 563/2 (8 + 1132)
= 563/2 × 1140
= 563 × 1140/2
= 641820/2 = 320910
Thus, the sum of all terms of the given even numbers from 8 to 1132 = 320910
And, the total number of terms = 563
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 1132
= 320910/563 = 570
Thus, the average of the given even numbers from 8 to 1132 = 570 Answer
Similar Questions
(1) What is the average of the first 1326 even numbers?
(2) Find the average of the first 2749 even numbers.
(3) Find the average of the first 3950 odd numbers.
(4) Find the average of odd numbers from 13 to 217
(5) Find the average of odd numbers from 13 to 717
(6) Find the average of even numbers from 8 to 432
(7) Find the average of odd numbers from 11 to 137
(8) What will be the average of the first 4312 odd numbers?
(9) Find the average of the first 2362 odd numbers.
(10) Find the average of the first 2001 odd numbers.