Question : Find the average of even numbers from 4 to 546
Correct Answer 275
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 546
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 546 are
4, 6, 8, . . . . 546
After observing the above list of the even numbers from 4 to 546 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 546 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 546
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 546
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 546
= 4 + 546/2
= 550/2 = 275
Thus, the average of the even numbers from 4 to 546 = 275 Answer
Method (2) to find the average of the even numbers from 4 to 546
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 546 are
4, 6, 8, . . . . 546
The even numbers from 4 to 546 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 546
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 546
546 = 4 + (n – 1) × 2
⇒ 546 = 4 + 2 n – 2
⇒ 546 = 4 – 2 + 2 n
⇒ 546 = 2 + 2 n
After transposing 2 to LHS
⇒ 546 – 2 = 2 n
⇒ 544 = 2 n
After rearranging the above expression
⇒ 2 n = 544
After transposing 2 to RHS
⇒ n = 544/2
⇒ n = 272
Thus, the number of terms of even numbers from 4 to 546 = 272
This means 546 is the 272th term.
Finding the sum of the given even numbers from 4 to 546
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 546
= 272/2 (4 + 546)
= 272/2 × 550
= 272 × 550/2
= 149600/2 = 74800
Thus, the sum of all terms of the given even numbers from 4 to 546 = 74800
And, the total number of terms = 272
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 546
= 74800/272 = 275
Thus, the average of the given even numbers from 4 to 546 = 275 Answer
Similar Questions
(1) Find the average of the first 2551 odd numbers.
(2) Find the average of the first 2665 odd numbers.
(3) Find the average of the first 3982 odd numbers.
(4) Find the average of the first 2106 odd numbers.
(5) What is the average of the first 1453 even numbers?
(6) Find the average of the first 549 odd numbers.
(7) Find the average of the first 348 odd numbers.
(8) Find the average of the first 3737 even numbers.
(9) Find the average of the first 464 odd numbers.
(10) Find the average of the first 3063 even numbers.