Question:
Find the average of even numbers from 10 to 1674
Correct Answer
842
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1674
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1674 are
10, 12, 14, . . . . 1674
After observing the above list of the even numbers from 10 to 1674 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1674 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1674
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1674
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 1674
= 10 + 1674/2
= 1684/2 = 842
Thus, the average of the even numbers from 10 to 1674 = 842 Answer
Method (2) to find the average of the even numbers from 10 to 1674
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1674 are
10, 12, 14, . . . . 1674
The even numbers from 10 to 1674 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1674
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 1674
1674 = 10 + (n – 1) × 2
⇒ 1674 = 10 + 2 n – 2
⇒ 1674 = 10 – 2 + 2 n
⇒ 1674 = 8 + 2 n
After transposing 8 to LHS
⇒ 1674 – 8 = 2 n
⇒ 1666 = 2 n
After rearranging the above expression
⇒ 2 n = 1666
After transposing 2 to RHS
⇒ n = 1666/2
⇒ n = 833
Thus, the number of terms of even numbers from 10 to 1674 = 833
This means 1674 is the 833th term.
Finding the sum of the given even numbers from 10 to 1674
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 1674
= 833/2 (10 + 1674)
= 833/2 × 1684
= 833 × 1684/2
= 1402772/2 = 701386
Thus, the sum of all terms of the given even numbers from 10 to 1674 = 701386
And, the total number of terms = 833
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 1674
= 701386/833 = 842
Thus, the average of the given even numbers from 10 to 1674 = 842 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 62
(2) Find the average of even numbers from 6 to 1006
(3) Find the average of the first 3064 odd numbers.
(4) What is the average of the first 808 even numbers?
(5) What is the average of the first 695 even numbers?
(6) Find the average of odd numbers from 11 to 1451
(7) Find the average of even numbers from 8 to 926
(8) Find the average of the first 4121 even numbers.
(9) Find the average of even numbers from 6 to 1698
(10) Find the average of odd numbers from 3 to 527