Question : Find the average of even numbers from 4 to 1682
Correct Answer 843
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1682
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1682 are
4, 6, 8, . . . . 1682
After observing the above list of the even numbers from 4 to 1682 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1682 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1682
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1682
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 4 to 1682
= 4 + 1682/2
= 1686/2 = 843
Thus, the average of the even numbers from 4 to 1682 = 843 Answer
Method (2) to find the average of the even numbers from 4 to 1682
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1682 are
4, 6, 8, . . . . 1682
The even numbers from 4 to 1682 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1682
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 4 to 1682
1682 = 4 + (n – 1) × 2
⇒ 1682 = 4 + 2 n – 2
⇒ 1682 = 4 – 2 + 2 n
⇒ 1682 = 2 + 2 n
After transposing 2 to LHS
⇒ 1682 – 2 = 2 n
⇒ 1680 = 2 n
After rearranging the above expression
⇒ 2 n = 1680
After transposing 2 to RHS
⇒ n = 1680/2
⇒ n = 840
Thus, the number of terms of even numbers from 4 to 1682 = 840
This means 1682 is the 840th term.
Finding the sum of the given even numbers from 4 to 1682
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 4 to 1682
= 840/2 (4 + 1682)
= 840/2 × 1686
= 840 × 1686/2
= 1416240/2 = 708120
Thus, the sum of all terms of the given even numbers from 4 to 1682 = 708120
And, the total number of terms = 840
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 4 to 1682
= 708120/840 = 843
Thus, the average of the given even numbers from 4 to 1682 = 843 Answer
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