Question : Find the average of even numbers from 4 to 384
Correct Answer 194
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 384
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 384 are
4, 6, 8, . . . . 384
After observing the above list of the even numbers from 4 to 384 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 384 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 384
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 384
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 384
= 4 + 384/2
= 388/2 = 194
Thus, the average of the even numbers from 4 to 384 = 194 Answer
Method (2) to find the average of the even numbers from 4 to 384
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 384 are
4, 6, 8, . . . . 384
The even numbers from 4 to 384 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 384
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 384
384 = 4 + (n – 1) × 2
⇒ 384 = 4 + 2 n – 2
⇒ 384 = 4 – 2 + 2 n
⇒ 384 = 2 + 2 n
After transposing 2 to LHS
⇒ 384 – 2 = 2 n
⇒ 382 = 2 n
After rearranging the above expression
⇒ 2 n = 382
After transposing 2 to RHS
⇒ n = 382/2
⇒ n = 191
Thus, the number of terms of even numbers from 4 to 384 = 191
This means 384 is the 191th term.
Finding the sum of the given even numbers from 4 to 384
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 384
= 191/2 (4 + 384)
= 191/2 × 388
= 191 × 388/2
= 74108/2 = 37054
Thus, the sum of all terms of the given even numbers from 4 to 384 = 37054
And, the total number of terms = 191
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 384
= 37054/191 = 194
Thus, the average of the given even numbers from 4 to 384 = 194 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 1002
(2) Find the average of even numbers from 12 to 1492
(3) Find the average of odd numbers from 3 to 737
(4) Find the average of odd numbers from 13 to 557
(5) Find the average of even numbers from 8 to 518
(6) Find the average of odd numbers from 11 to 221
(7) Find the average of the first 2860 even numbers.
(8) Find the average of odd numbers from 13 to 623
(9) Find the average of even numbers from 10 to 888
(10) Find the average of the first 2065 even numbers.