Question:
Find the average of even numbers from 12 to 1496
Correct Answer
754
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1496
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1496 are
12, 14, 16, . . . . 1496
After observing the above list of the even numbers from 12 to 1496 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1496 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1496
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1496
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 1496
= 12 + 1496/2
= 1508/2 = 754
Thus, the average of the even numbers from 12 to 1496 = 754 Answer
Method (2) to find the average of the even numbers from 12 to 1496
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1496 are
12, 14, 16, . . . . 1496
The even numbers from 12 to 1496 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1496
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 1496
1496 = 12 + (n – 1) × 2
⇒ 1496 = 12 + 2 n – 2
⇒ 1496 = 12 – 2 + 2 n
⇒ 1496 = 10 + 2 n
After transposing 10 to LHS
⇒ 1496 – 10 = 2 n
⇒ 1486 = 2 n
After rearranging the above expression
⇒ 2 n = 1486
After transposing 2 to RHS
⇒ n = 1486/2
⇒ n = 743
Thus, the number of terms of even numbers from 12 to 1496 = 743
This means 1496 is the 743th term.
Finding the sum of the given even numbers from 12 to 1496
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 1496
= 743/2 (12 + 1496)
= 743/2 × 1508
= 743 × 1508/2
= 1120444/2 = 560222
Thus, the sum of all terms of the given even numbers from 12 to 1496 = 560222
And, the total number of terms = 743
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 1496
= 560222/743 = 754
Thus, the average of the given even numbers from 12 to 1496 = 754 Answer
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