Question:
Find the average of even numbers from 12 to 1794
Correct Answer
903
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1794
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1794 are
12, 14, 16, . . . . 1794
After observing the above list of the even numbers from 12 to 1794 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1794 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1794
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1794
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 12 to 1794
= 12 + 1794/2
= 1806/2 = 903
Thus, the average of the even numbers from 12 to 1794 = 903 Answer
Method (2) to find the average of the even numbers from 12 to 1794
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1794 are
12, 14, 16, . . . . 1794
The even numbers from 12 to 1794 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1794
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 12 to 1794
1794 = 12 + (n – 1) × 2
⇒ 1794 = 12 + 2 n – 2
⇒ 1794 = 12 – 2 + 2 n
⇒ 1794 = 10 + 2 n
After transposing 10 to LHS
⇒ 1794 – 10 = 2 n
⇒ 1784 = 2 n
After rearranging the above expression
⇒ 2 n = 1784
After transposing 2 to RHS
⇒ n = 1784/2
⇒ n = 892
Thus, the number of terms of even numbers from 12 to 1794 = 892
This means 1794 is the 892th term.
Finding the sum of the given even numbers from 12 to 1794
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 12 to 1794
= 892/2 (12 + 1794)
= 892/2 × 1806
= 892 × 1806/2
= 1610952/2 = 805476
Thus, the sum of all terms of the given even numbers from 12 to 1794 = 805476
And, the total number of terms = 892
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 12 to 1794
= 805476/892 = 903
Thus, the average of the given even numbers from 12 to 1794 = 903 Answer
Similar Questions
(1) Find the average of odd numbers from 13 to 673
(2) Find the average of odd numbers from 5 to 1245
(3) Find the average of odd numbers from 15 to 1023
(4) Find the average of the first 3144 even numbers.
(5) Find the average of the first 2619 even numbers.
(6) Find the average of even numbers from 12 to 642
(7) Find the average of the first 3544 odd numbers.
(8) Find the average of even numbers from 4 to 724
(9) Find the average of the first 931 odd numbers.
(10) Find the average of odd numbers from 3 to 981