Question:
Find the average of even numbers from 12 to 424
Correct Answer
218
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 424
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 424 are
12, 14, 16, . . . . 424
After observing the above list of the even numbers from 12 to 424 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 424 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 424
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 424
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 424
= 12 + 424/2
= 436/2 = 218
Thus, the average of the even numbers from 12 to 424 = 218 Answer
Method (2) to find the average of the even numbers from 12 to 424
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 424 are
12, 14, 16, . . . . 424
The even numbers from 12 to 424 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 424
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 424
424 = 12 + (n – 1) × 2
⇒ 424 = 12 + 2 n – 2
⇒ 424 = 12 – 2 + 2 n
⇒ 424 = 10 + 2 n
After transposing 10 to LHS
⇒ 424 – 10 = 2 n
⇒ 414 = 2 n
After rearranging the above expression
⇒ 2 n = 414
After transposing 2 to RHS
⇒ n = 414/2
⇒ n = 207
Thus, the number of terms of even numbers from 12 to 424 = 207
This means 424 is the 207th term.
Finding the sum of the given even numbers from 12 to 424
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 424
= 207/2 (12 + 424)
= 207/2 × 436
= 207 × 436/2
= 90252/2 = 45126
Thus, the sum of all terms of the given even numbers from 12 to 424 = 45126
And, the total number of terms = 207
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 424
= 45126/207 = 218
Thus, the average of the given even numbers from 12 to 424 = 218 Answer
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