Question:
Find the average of even numbers from 10 to 1750
Correct Answer
880
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1750
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1750 are
10, 12, 14, . . . . 1750
After observing the above list of the even numbers from 10 to 1750 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1750 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1750
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1750
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 1750
= 10 + 1750/2
= 1760/2 = 880
Thus, the average of the even numbers from 10 to 1750 = 880 Answer
Method (2) to find the average of the even numbers from 10 to 1750
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1750 are
10, 12, 14, . . . . 1750
The even numbers from 10 to 1750 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1750
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 1750
1750 = 10 + (n – 1) × 2
⇒ 1750 = 10 + 2 n – 2
⇒ 1750 = 10 – 2 + 2 n
⇒ 1750 = 8 + 2 n
After transposing 8 to LHS
⇒ 1750 – 8 = 2 n
⇒ 1742 = 2 n
After rearranging the above expression
⇒ 2 n = 1742
After transposing 2 to RHS
⇒ n = 1742/2
⇒ n = 871
Thus, the number of terms of even numbers from 10 to 1750 = 871
This means 1750 is the 871th term.
Finding the sum of the given even numbers from 10 to 1750
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 1750
= 871/2 (10 + 1750)
= 871/2 × 1760
= 871 × 1760/2
= 1532960/2 = 766480
Thus, the sum of all terms of the given even numbers from 10 to 1750 = 766480
And, the total number of terms = 871
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 1750
= 766480/871 = 880
Thus, the average of the given even numbers from 10 to 1750 = 880 Answer
Similar Questions
(1) Find the average of even numbers from 4 to 1474
(2) Find the average of even numbers from 4 to 1308
(3) What will be the average of the first 4843 odd numbers?
(4) Find the average of odd numbers from 13 to 705
(5) Find the average of the first 2733 odd numbers.
(6) Find the average of the first 1028 odd numbers.
(7) Find the average of the first 2646 odd numbers.
(8) Find the average of odd numbers from 7 to 787
(9) What will be the average of the first 4247 odd numbers?
(10) Find the average of the first 4755 even numbers.