Question : Find the average of even numbers from 4 to 1756
Correct Answer 880
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1756
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1756 are
4, 6, 8, . . . . 1756
After observing the above list of the even numbers from 4 to 1756 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1756 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1756
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1756
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 4 to 1756
= 4 + 1756/2
= 1760/2 = 880
Thus, the average of the even numbers from 4 to 1756 = 880 Answer
Method (2) to find the average of the even numbers from 4 to 1756
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1756 are
4, 6, 8, . . . . 1756
The even numbers from 4 to 1756 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1756
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 4 to 1756
1756 = 4 + (n – 1) × 2
⇒ 1756 = 4 + 2 n – 2
⇒ 1756 = 4 – 2 + 2 n
⇒ 1756 = 2 + 2 n
After transposing 2 to LHS
⇒ 1756 – 2 = 2 n
⇒ 1754 = 2 n
After rearranging the above expression
⇒ 2 n = 1754
After transposing 2 to RHS
⇒ n = 1754/2
⇒ n = 877
Thus, the number of terms of even numbers from 4 to 1756 = 877
This means 1756 is the 877th term.
Finding the sum of the given even numbers from 4 to 1756
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 4 to 1756
= 877/2 (4 + 1756)
= 877/2 × 1760
= 877 × 1760/2
= 1543520/2 = 771760
Thus, the sum of all terms of the given even numbers from 4 to 1756 = 771760
And, the total number of terms = 877
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 4 to 1756
= 771760/877 = 880
Thus, the average of the given even numbers from 4 to 1756 = 880 Answer
Similar Questions
(1) What is the average of the first 1633 even numbers?
(2) Find the average of odd numbers from 7 to 151
(3) Find the average of odd numbers from 13 to 1221
(4) Find the average of odd numbers from 15 to 681
(5) What is the average of the first 1967 even numbers?
(6) Find the average of the first 2346 odd numbers.
(7) Find the average of even numbers from 12 to 726
(8) What will be the average of the first 4198 odd numbers?
(9) What will be the average of the first 4148 odd numbers?
(10) Find the average of the first 3471 even numbers.