Question:
Find the average of even numbers from 10 to 1710
Correct Answer
860
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1710
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1710 are
10, 12, 14, . . . . 1710
After observing the above list of the even numbers from 10 to 1710 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1710 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1710
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1710
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 1710
= 10 + 1710/2
= 1720/2 = 860
Thus, the average of the even numbers from 10 to 1710 = 860 Answer
Method (2) to find the average of the even numbers from 10 to 1710
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1710 are
10, 12, 14, . . . . 1710
The even numbers from 10 to 1710 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1710
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 1710
1710 = 10 + (n – 1) × 2
⇒ 1710 = 10 + 2 n – 2
⇒ 1710 = 10 – 2 + 2 n
⇒ 1710 = 8 + 2 n
After transposing 8 to LHS
⇒ 1710 – 8 = 2 n
⇒ 1702 = 2 n
After rearranging the above expression
⇒ 2 n = 1702
After transposing 2 to RHS
⇒ n = 1702/2
⇒ n = 851
Thus, the number of terms of even numbers from 10 to 1710 = 851
This means 1710 is the 851th term.
Finding the sum of the given even numbers from 10 to 1710
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 1710
= 851/2 (10 + 1710)
= 851/2 × 1720
= 851 × 1720/2
= 1463720/2 = 731860
Thus, the sum of all terms of the given even numbers from 10 to 1710 = 731860
And, the total number of terms = 851
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 1710
= 731860/851 = 860
Thus, the average of the given even numbers from 10 to 1710 = 860 Answer
Similar Questions
(1) Find the average of the first 228 odd numbers.
(2) Find the average of odd numbers from 3 to 1425
(3) Find the average of even numbers from 10 to 1674
(4) Find the average of even numbers from 8 to 528
(5) Find the average of odd numbers from 5 to 1411
(6) Find the average of even numbers from 8 to 1308
(7) What is the average of the first 620 even numbers?
(8) Find the average of the first 2034 odd numbers.
(9) Find the average of the first 2807 even numbers.
(10) What is the average of the first 1409 even numbers?