Question:
Find the average of even numbers from 4 to 1538
Correct Answer
771
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1538
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1538 are
4, 6, 8, . . . . 1538
After observing the above list of the even numbers from 4 to 1538 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1538 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1538
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1538
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 1538
= 4 + 1538/2
= 1542/2 = 771
Thus, the average of the even numbers from 4 to 1538 = 771 Answer
Method (2) to find the average of the even numbers from 4 to 1538
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1538 are
4, 6, 8, . . . . 1538
The even numbers from 4 to 1538 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1538
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 1538
1538 = 4 + (n – 1) × 2
⇒ 1538 = 4 + 2 n – 2
⇒ 1538 = 4 – 2 + 2 n
⇒ 1538 = 2 + 2 n
After transposing 2 to LHS
⇒ 1538 – 2 = 2 n
⇒ 1536 = 2 n
After rearranging the above expression
⇒ 2 n = 1536
After transposing 2 to RHS
⇒ n = 1536/2
⇒ n = 768
Thus, the number of terms of even numbers from 4 to 1538 = 768
This means 1538 is the 768th term.
Finding the sum of the given even numbers from 4 to 1538
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 1538
= 768/2 (4 + 1538)
= 768/2 × 1542
= 768 × 1542/2
= 1184256/2 = 592128
Thus, the sum of all terms of the given even numbers from 4 to 1538 = 592128
And, the total number of terms = 768
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 1538
= 592128/768 = 771
Thus, the average of the given even numbers from 4 to 1538 = 771 Answer
Similar Questions
(1) What is the average of the first 289 even numbers?
(2) Find the average of the first 3503 odd numbers.
(3) Find the average of the first 4857 even numbers.
(4) Find the average of the first 1598 odd numbers.
(5) Find the average of the first 3702 odd numbers.
(6) Find the average of the first 1914 odd numbers.
(7) What is the average of the first 1207 even numbers?
(8) What is the average of the first 177 even numbers?
(9) Find the average of the first 1543 odd numbers.
(10) Find the average of the first 3309 even numbers.