Question:
Find the average of even numbers from 4 to 778
Correct Answer
391
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 778
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 778 are
4, 6, 8, . . . . 778
After observing the above list of the even numbers from 4 to 778 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 778 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 778
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 778
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 778
= 4 + 778/2
= 782/2 = 391
Thus, the average of the even numbers from 4 to 778 = 391 Answer
Method (2) to find the average of the even numbers from 4 to 778
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 778 are
4, 6, 8, . . . . 778
The even numbers from 4 to 778 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 778
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 778
778 = 4 + (n – 1) × 2
⇒ 778 = 4 + 2 n – 2
⇒ 778 = 4 – 2 + 2 n
⇒ 778 = 2 + 2 n
After transposing 2 to LHS
⇒ 778 – 2 = 2 n
⇒ 776 = 2 n
After rearranging the above expression
⇒ 2 n = 776
After transposing 2 to RHS
⇒ n = 776/2
⇒ n = 388
Thus, the number of terms of even numbers from 4 to 778 = 388
This means 778 is the 388th term.
Finding the sum of the given even numbers from 4 to 778
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 778
= 388/2 (4 + 778)
= 388/2 × 782
= 388 × 782/2
= 303416/2 = 151708
Thus, the sum of all terms of the given even numbers from 4 to 778 = 151708
And, the total number of terms = 388
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 778
= 151708/388 = 391
Thus, the average of the given even numbers from 4 to 778 = 391 Answer
Similar Questions
(1) Find the average of the first 3316 even numbers.
(2) Find the average of odd numbers from 13 to 433
(3) Find the average of even numbers from 10 to 1650
(4) Find the average of odd numbers from 13 to 1447
(5) Find the average of the first 3243 odd numbers.
(6) Find the average of the first 4448 even numbers.
(7) Find the average of the first 3793 even numbers.
(8) Find the average of even numbers from 10 to 1152
(9) Find the average of the first 3668 even numbers.
(10) Find the average of the first 3932 odd numbers.