Question:
Find the average of even numbers from 10 to 840
Correct Answer
425
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 840
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 840 are
10, 12, 14, . . . . 840
After observing the above list of the even numbers from 10 to 840 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 840 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 840
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 840
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 840
= 10 + 840/2
= 850/2 = 425
Thus, the average of the even numbers from 10 to 840 = 425 Answer
Method (2) to find the average of the even numbers from 10 to 840
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 840 are
10, 12, 14, . . . . 840
The even numbers from 10 to 840 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 840
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 840
840 = 10 + (n – 1) × 2
⇒ 840 = 10 + 2 n – 2
⇒ 840 = 10 – 2 + 2 n
⇒ 840 = 8 + 2 n
After transposing 8 to LHS
⇒ 840 – 8 = 2 n
⇒ 832 = 2 n
After rearranging the above expression
⇒ 2 n = 832
After transposing 2 to RHS
⇒ n = 832/2
⇒ n = 416
Thus, the number of terms of even numbers from 10 to 840 = 416
This means 840 is the 416th term.
Finding the sum of the given even numbers from 10 to 840
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 840
= 416/2 (10 + 840)
= 416/2 × 850
= 416 × 850/2
= 353600/2 = 176800
Thus, the sum of all terms of the given even numbers from 10 to 840 = 176800
And, the total number of terms = 416
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 840
= 176800/416 = 425
Thus, the average of the given even numbers from 10 to 840 = 425 Answer
Similar Questions
(1) Find the average of the first 3782 odd numbers.
(2) Find the average of odd numbers from 9 to 231
(3) Find the average of the first 3064 odd numbers.
(4) Find the average of odd numbers from 5 to 1267
(5) Find the average of even numbers from 6 to 824
(6) Find the average of even numbers from 10 to 450
(7) Find the average of even numbers from 10 to 320
(8) Find the average of even numbers from 6 to 1280
(9) Find the average of even numbers from 12 to 846
(10) What will be the average of the first 4820 odd numbers?