Question : Find the average of even numbers from 4 to 1940
Correct Answer 972
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 4 to 1940
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 4 to 1940 are
4, 6, 8, . . . . 1940
After observing the above list of the even numbers from 4 to 1940 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 4 to 1940 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 4 to 1940
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1940
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 1940
= 4 + 1940/2
= 1944/2 = 972
Thus, the average of the even numbers from 4 to 1940 = 972 Answer
Method (2) to find the average of the even numbers from 4 to 1940
Finding the average of given continuous even numbers after finding their sum
The even numbers from 4 to 1940 are
4, 6, 8, . . . . 1940
The even numbers from 4 to 1940 form an Arithmetic Series in which
The First Term (a) = 4
The Common Difference (d) = 2
And the last term (ℓ) = 1940
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 1940
1940 = 4 + (n – 1) × 2
⇒ 1940 = 4 + 2 n – 2
⇒ 1940 = 4 – 2 + 2 n
⇒ 1940 = 2 + 2 n
After transposing 2 to LHS
⇒ 1940 – 2 = 2 n
⇒ 1938 = 2 n
After rearranging the above expression
⇒ 2 n = 1938
After transposing 2 to RHS
⇒ n = 1938/2
⇒ n = 969
Thus, the number of terms of even numbers from 4 to 1940 = 969
This means 1940 is the 969th term.
Finding the sum of the given even numbers from 4 to 1940
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 1940
= 969/2 (4 + 1940)
= 969/2 × 1944
= 969 × 1944/2
= 1883736/2 = 941868
Thus, the sum of all terms of the given even numbers from 4 to 1940 = 941868
And, the total number of terms = 969
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 1940
= 941868/969 = 972
Thus, the average of the given even numbers from 4 to 1940 = 972 Answer
Similar Questions
(1) Find the average of the first 3011 even numbers.
(2) Find the average of odd numbers from 9 to 419
(3) Find the average of the first 2532 odd numbers.
(4) Find the average of odd numbers from 11 to 1009
(5) Find the average of odd numbers from 11 to 1143
(6) Find the average of odd numbers from 7 to 503
(7) Find the average of odd numbers from 11 to 853
(8) Find the average of the first 4065 even numbers.
(9) Find the average of odd numbers from 11 to 1445
(10) Find the average of the first 2084 odd numbers.