Question:
Find the average of even numbers from 12 to 506
Correct Answer
259
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 506
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 506 are
12, 14, 16, . . . . 506
After observing the above list of the even numbers from 12 to 506 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 506 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 506
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 506
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 506
= 12 + 506/2
= 518/2 = 259
Thus, the average of the even numbers from 12 to 506 = 259 Answer
Method (2) to find the average of the even numbers from 12 to 506
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 506 are
12, 14, 16, . . . . 506
The even numbers from 12 to 506 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 506
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 506
506 = 12 + (n – 1) × 2
⇒ 506 = 12 + 2 n – 2
⇒ 506 = 12 – 2 + 2 n
⇒ 506 = 10 + 2 n
After transposing 10 to LHS
⇒ 506 – 10 = 2 n
⇒ 496 = 2 n
After rearranging the above expression
⇒ 2 n = 496
After transposing 2 to RHS
⇒ n = 496/2
⇒ n = 248
Thus, the number of terms of even numbers from 12 to 506 = 248
This means 506 is the 248th term.
Finding the sum of the given even numbers from 12 to 506
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 506
= 248/2 (12 + 506)
= 248/2 × 518
= 248 × 518/2
= 128464/2 = 64232
Thus, the sum of all terms of the given even numbers from 12 to 506 = 64232
And, the total number of terms = 248
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 506
= 64232/248 = 259
Thus, the average of the given even numbers from 12 to 506 = 259 Answer
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