Question:
Find the average of even numbers from 12 to 1666
Correct Answer
839
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1666
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1666 are
12, 14, 16, . . . . 1666
After observing the above list of the even numbers from 12 to 1666 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1666 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1666
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1666
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 1666
= 12 + 1666/2
= 1678/2 = 839
Thus, the average of the even numbers from 12 to 1666 = 839 Answer
Method (2) to find the average of the even numbers from 12 to 1666
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1666 are
12, 14, 16, . . . . 1666
The even numbers from 12 to 1666 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1666
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 1666
1666 = 12 + (n – 1) × 2
⇒ 1666 = 12 + 2 n – 2
⇒ 1666 = 12 – 2 + 2 n
⇒ 1666 = 10 + 2 n
After transposing 10 to LHS
⇒ 1666 – 10 = 2 n
⇒ 1656 = 2 n
After rearranging the above expression
⇒ 2 n = 1656
After transposing 2 to RHS
⇒ n = 1656/2
⇒ n = 828
Thus, the number of terms of even numbers from 12 to 1666 = 828
This means 1666 is the 828th term.
Finding the sum of the given even numbers from 12 to 1666
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 1666
= 828/2 (12 + 1666)
= 828/2 × 1678
= 828 × 1678/2
= 1389384/2 = 694692
Thus, the sum of all terms of the given even numbers from 12 to 1666 = 694692
And, the total number of terms = 828
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 1666
= 694692/828 = 839
Thus, the average of the given even numbers from 12 to 1666 = 839 Answer
Similar Questions
(1) What will be the average of the first 4311 odd numbers?
(2) Find the average of even numbers from 6 to 718
(3) Find the average of even numbers from 10 to 2000
(4) Find the average of odd numbers from 11 to 1255
(5) Find the average of even numbers from 6 to 1350
(6) Find the average of even numbers from 6 to 1186
(7) Find the average of the first 1215 odd numbers.
(8) Find the average of even numbers from 12 to 466
(9) Find the average of even numbers from 6 to 1626
(10) Find the average of the first 599 odd numbers.