Question:
Find the average of even numbers from 12 to 1428
Correct Answer
720
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1428
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1428 are
12, 14, 16, . . . . 1428
After observing the above list of the even numbers from 12 to 1428 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1428 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1428
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1428
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 1428
= 12 + 1428/2
= 1440/2 = 720
Thus, the average of the even numbers from 12 to 1428 = 720 Answer
Method (2) to find the average of the even numbers from 12 to 1428
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1428 are
12, 14, 16, . . . . 1428
The even numbers from 12 to 1428 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1428
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 1428
1428 = 12 + (n – 1) × 2
⇒ 1428 = 12 + 2 n – 2
⇒ 1428 = 12 – 2 + 2 n
⇒ 1428 = 10 + 2 n
After transposing 10 to LHS
⇒ 1428 – 10 = 2 n
⇒ 1418 = 2 n
After rearranging the above expression
⇒ 2 n = 1418
After transposing 2 to RHS
⇒ n = 1418/2
⇒ n = 709
Thus, the number of terms of even numbers from 12 to 1428 = 709
This means 1428 is the 709th term.
Finding the sum of the given even numbers from 12 to 1428
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 1428
= 709/2 (12 + 1428)
= 709/2 × 1440
= 709 × 1440/2
= 1020960/2 = 510480
Thus, the sum of all terms of the given even numbers from 12 to 1428 = 510480
And, the total number of terms = 709
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 1428
= 510480/709 = 720
Thus, the average of the given even numbers from 12 to 1428 = 720 Answer
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