Question:
Find the average of even numbers from 10 to 1452
Correct Answer
731
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1452
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1452 are
10, 12, 14, . . . . 1452
After observing the above list of the even numbers from 10 to 1452 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1452 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1452
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1452
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 1452
= 10 + 1452/2
= 1462/2 = 731
Thus, the average of the even numbers from 10 to 1452 = 731 Answer
Method (2) to find the average of the even numbers from 10 to 1452
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1452 are
10, 12, 14, . . . . 1452
The even numbers from 10 to 1452 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1452
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 1452
1452 = 10 + (n – 1) × 2
⇒ 1452 = 10 + 2 n – 2
⇒ 1452 = 10 – 2 + 2 n
⇒ 1452 = 8 + 2 n
After transposing 8 to LHS
⇒ 1452 – 8 = 2 n
⇒ 1444 = 2 n
After rearranging the above expression
⇒ 2 n = 1444
After transposing 2 to RHS
⇒ n = 1444/2
⇒ n = 722
Thus, the number of terms of even numbers from 10 to 1452 = 722
This means 1452 is the 722th term.
Finding the sum of the given even numbers from 10 to 1452
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 1452
= 722/2 (10 + 1452)
= 722/2 × 1462
= 722 × 1462/2
= 1055564/2 = 527782
Thus, the sum of all terms of the given even numbers from 10 to 1452 = 527782
And, the total number of terms = 722
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 1452
= 527782/722 = 731
Thus, the average of the given even numbers from 10 to 1452 = 731 Answer
Similar Questions
(1) What will be the average of the first 4878 odd numbers?
(2) Find the average of even numbers from 10 to 464
(3) Find the average of the first 2126 even numbers.
(4) Find the average of even numbers from 10 to 1530
(5) Find the average of the first 2813 even numbers.
(6) What will be the average of the first 4905 odd numbers?
(7) What will be the average of the first 4337 odd numbers?
(8) Find the average of the first 2620 even numbers.
(9) Find the average of even numbers from 10 to 1350
(10) Find the average of even numbers from 8 to 1332