Question:
Find the average of even numbers from 10 to 1784
Correct Answer
897
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1784
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1784 are
10, 12, 14, . . . . 1784
After observing the above list of the even numbers from 10 to 1784 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1784 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1784
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1784
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 10 to 1784
= 10 + 1784/2
= 1794/2 = 897
Thus, the average of the even numbers from 10 to 1784 = 897 Answer
Method (2) to find the average of the even numbers from 10 to 1784
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1784 are
10, 12, 14, . . . . 1784
The even numbers from 10 to 1784 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1784
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 10 to 1784
1784 = 10 + (n – 1) × 2
⇒ 1784 = 10 + 2 n – 2
⇒ 1784 = 10 – 2 + 2 n
⇒ 1784 = 8 + 2 n
After transposing 8 to LHS
⇒ 1784 – 8 = 2 n
⇒ 1776 = 2 n
After rearranging the above expression
⇒ 2 n = 1776
After transposing 2 to RHS
⇒ n = 1776/2
⇒ n = 888
Thus, the number of terms of even numbers from 10 to 1784 = 888
This means 1784 is the 888th term.
Finding the sum of the given even numbers from 10 to 1784
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 10 to 1784
= 888/2 (10 + 1784)
= 888/2 × 1794
= 888 × 1794/2
= 1593072/2 = 796536
Thus, the sum of all terms of the given even numbers from 10 to 1784 = 796536
And, the total number of terms = 888
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 10 to 1784
= 796536/888 = 897
Thus, the average of the given even numbers from 10 to 1784 = 897 Answer
Similar Questions
(1) What is the average of the first 973 even numbers?
(2) What is the average of the first 483 even numbers?
(3) Find the average of odd numbers from 5 to 539
(4) Find the average of even numbers from 6 to 1798
(5) What is the average of the first 1880 even numbers?
(6) Find the average of even numbers from 6 to 250
(7) Find the average of odd numbers from 7 to 167
(8) Find the average of odd numbers from 13 to 1467
(9) Find the average of odd numbers from 3 to 1363
(10) Find the average of odd numbers from 3 to 749