Question:
Find the average of even numbers from 8 to 580
Correct Answer
294
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 580
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 580 are
8, 10, 12, . . . . 580
After observing the above list of the even numbers from 8 to 580 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 580 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 580
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 580
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 8 to 580
= 8 + 580/2
= 588/2 = 294
Thus, the average of the even numbers from 8 to 580 = 294 Answer
Method (2) to find the average of the even numbers from 8 to 580
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 580 are
8, 10, 12, . . . . 580
The even numbers from 8 to 580 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 580
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 8 to 580
580 = 8 + (n – 1) × 2
⇒ 580 = 8 + 2 n – 2
⇒ 580 = 8 – 2 + 2 n
⇒ 580 = 6 + 2 n
After transposing 6 to LHS
⇒ 580 – 6 = 2 n
⇒ 574 = 2 n
After rearranging the above expression
⇒ 2 n = 574
After transposing 2 to RHS
⇒ n = 574/2
⇒ n = 287
Thus, the number of terms of even numbers from 8 to 580 = 287
This means 580 is the 287th term.
Finding the sum of the given even numbers from 8 to 580
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 8 to 580
= 287/2 (8 + 580)
= 287/2 × 588
= 287 × 588/2
= 168756/2 = 84378
Thus, the sum of all terms of the given even numbers from 8 to 580 = 84378
And, the total number of terms = 287
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 8 to 580
= 84378/287 = 294
Thus, the average of the given even numbers from 8 to 580 = 294 Answer
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