Question : Find the average of even numbers from 12 to 582
Correct Answer 297
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 582
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 582 are
12, 14, 16, . . . . 582
After observing the above list of the even numbers from 12 to 582 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 582 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 582
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 582
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 12 to 582
= 12 + 582/2
= 594/2 = 297
Thus, the average of the even numbers from 12 to 582 = 297 Answer
Method (2) to find the average of the even numbers from 12 to 582
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 582 are
12, 14, 16, . . . . 582
The even numbers from 12 to 582 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 582
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 12 to 582
582 = 12 + (n – 1) × 2
⇒ 582 = 12 + 2 n – 2
⇒ 582 = 12 – 2 + 2 n
⇒ 582 = 10 + 2 n
After transposing 10 to LHS
⇒ 582 – 10 = 2 n
⇒ 572 = 2 n
After rearranging the above expression
⇒ 2 n = 572
After transposing 2 to RHS
⇒ n = 572/2
⇒ n = 286
Thus, the number of terms of even numbers from 12 to 582 = 286
This means 582 is the 286th term.
Finding the sum of the given even numbers from 12 to 582
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 12 to 582
= 286/2 (12 + 582)
= 286/2 × 594
= 286 × 594/2
= 169884/2 = 84942
Thus, the sum of all terms of the given even numbers from 12 to 582 = 84942
And, the total number of terms = 286
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 12 to 582
= 84942/286 = 297
Thus, the average of the given even numbers from 12 to 582 = 297 Answer
Similar Questions
(1) Find the average of odd numbers from 5 to 97
(2) Find the average of the first 2298 odd numbers.
(3) Find the average of odd numbers from 7 to 1451
(4) Find the average of odd numbers from 13 to 1247
(5) Find the average of odd numbers from 9 to 1359
(6) Find the average of even numbers from 12 to 452
(7) Find the average of the first 3725 even numbers.
(8) Find the average of even numbers from 8 to 714
(9) Find the average of odd numbers from 9 to 317
(10) Find the average of the first 4767 even numbers.