Question : Find the average of even numbers from 6 to 464
Correct Answer 235
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 464
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 464 are
6, 8, 10, . . . . 464
After observing the above list of the even numbers from 6 to 464 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 464 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 464
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 464
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 6 to 464
= 6 + 464/2
= 470/2 = 235
Thus, the average of the even numbers from 6 to 464 = 235 Answer
Method (2) to find the average of the even numbers from 6 to 464
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 464 are
6, 8, 10, . . . . 464
The even numbers from 6 to 464 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 464
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 6 to 464
464 = 6 + (n – 1) × 2
⇒ 464 = 6 + 2 n – 2
⇒ 464 = 6 – 2 + 2 n
⇒ 464 = 4 + 2 n
After transposing 4 to LHS
⇒ 464 – 4 = 2 n
⇒ 460 = 2 n
After rearranging the above expression
⇒ 2 n = 460
After transposing 2 to RHS
⇒ n = 460/2
⇒ n = 230
Thus, the number of terms of even numbers from 6 to 464 = 230
This means 464 is the 230th term.
Finding the sum of the given even numbers from 6 to 464
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 6 to 464
= 230/2 (6 + 464)
= 230/2 × 470
= 230 × 470/2
= 108100/2 = 54050
Thus, the sum of all terms of the given even numbers from 6 to 464 = 54050
And, the total number of terms = 230
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 6 to 464
= 54050/230 = 235
Thus, the average of the given even numbers from 6 to 464 = 235 Answer
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