Question:
Find the average of even numbers from 4 to 1386
Correct Answer
695
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1386
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1386 are
4, 6, 8, . . . . 1386
After observing the above list of the even numbers from 4 to 1386 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1386 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1386
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1386
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 4 to 1386
= 4 + 1386/2
= 1390/2 = 695
Thus, the average of the even numbers from 4 to 1386 = 695 Answer
Method (2) to find the average of the even numbers from 4 to 1386
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1386 are
4, 6, 8, . . . . 1386
The even numbers from 4 to 1386 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1386
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 4 to 1386
1386 = 4 + (n – 1) × 2
⇒ 1386 = 4 + 2 n – 2
⇒ 1386 = 4 – 2 + 2 n
⇒ 1386 = 2 + 2 n
After transposing 2 to LHS
⇒ 1386 – 2 = 2 n
⇒ 1384 = 2 n
After rearranging the above expression
⇒ 2 n = 1384
After transposing 2 to RHS
⇒ n = 1384/2
⇒ n = 692
Thus, the number of terms of even numbers from 4 to 1386 = 692
This means 1386 is the 692th term.
Finding the sum of the given even numbers from 4 to 1386
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 4 to 1386
= 692/2 (4 + 1386)
= 692/2 × 1390
= 692 × 1390/2
= 961880/2 = 480940
Thus, the sum of all terms of the given even numbers from 4 to 1386 = 480940
And, the total number of terms = 692
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 4 to 1386
= 480940/692 = 695
Thus, the average of the given even numbers from 4 to 1386 = 695 Answer
Similar Questions
(1) Find the average of the first 3214 odd numbers.
(2) Find the average of odd numbers from 13 to 1479
(3) What will be the average of the first 4378 odd numbers?
(4) Find the average of even numbers from 6 to 1712
(5) Find the average of even numbers from 10 to 1042
(6) What is the average of the first 892 even numbers?
(7) Find the average of even numbers from 4 to 582
(8) Find the average of even numbers from 8 to 854
(9) Find the average of the first 1726 odd numbers.
(10) Find the average of even numbers from 10 to 1054