Question:
Find the average of even numbers from 10 to 1870
Correct Answer
940
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1870
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1870 are
10, 12, 14, . . . . 1870
After observing the above list of the even numbers from 10 to 1870 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1870 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1870
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1870
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 1870
= 10 + 1870/2
= 1880/2 = 940
Thus, the average of the even numbers from 10 to 1870 = 940 Answer
Method (2) to find the average of the even numbers from 10 to 1870
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1870 are
10, 12, 14, . . . . 1870
The even numbers from 10 to 1870 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1870
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 1870
1870 = 10 + (n – 1) × 2
⇒ 1870 = 10 + 2 n – 2
⇒ 1870 = 10 – 2 + 2 n
⇒ 1870 = 8 + 2 n
After transposing 8 to LHS
⇒ 1870 – 8 = 2 n
⇒ 1862 = 2 n
After rearranging the above expression
⇒ 2 n = 1862
After transposing 2 to RHS
⇒ n = 1862/2
⇒ n = 931
Thus, the number of terms of even numbers from 10 to 1870 = 931
This means 1870 is the 931th term.
Finding the sum of the given even numbers from 10 to 1870
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 1870
= 931/2 (10 + 1870)
= 931/2 × 1880
= 931 × 1880/2
= 1750280/2 = 875140
Thus, the sum of all terms of the given even numbers from 10 to 1870 = 875140
And, the total number of terms = 931
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 1870
= 875140/931 = 940
Thus, the average of the given even numbers from 10 to 1870 = 940 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 1203
(2) Find the average of odd numbers from 9 to 579
(3) Find the average of even numbers from 4 to 1274
(4) Find the average of odd numbers from 3 to 1115
(5) Find the average of even numbers from 6 to 1304
(6) Find the average of the first 3080 odd numbers.
(7) Find the average of the first 516 odd numbers.
(8) Find the average of the first 1362 odd numbers.
(9) Find the average of the first 1530 odd numbers.
(10) Find the average of the first 394 odd numbers.