Question : Find the average of even numbers from 12 to 1964
Correct Answer 988
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1964
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1964 are
12, 14, 16, . . . . 1964
After observing the above list of the even numbers from 12 to 1964 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1964 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1964
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1964
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 1964
= 12 + 1964/2
= 1976/2 = 988
Thus, the average of the even numbers from 12 to 1964 = 988 Answer
Method (2) to find the average of the even numbers from 12 to 1964
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1964 are
12, 14, 16, . . . . 1964
The even numbers from 12 to 1964 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1964
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 1964
1964 = 12 + (n – 1) × 2
⇒ 1964 = 12 + 2 n – 2
⇒ 1964 = 12 – 2 + 2 n
⇒ 1964 = 10 + 2 n
After transposing 10 to LHS
⇒ 1964 – 10 = 2 n
⇒ 1954 = 2 n
After rearranging the above expression
⇒ 2 n = 1954
After transposing 2 to RHS
⇒ n = 1954/2
⇒ n = 977
Thus, the number of terms of even numbers from 12 to 1964 = 977
This means 1964 is the 977th term.
Finding the sum of the given even numbers from 12 to 1964
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 1964
= 977/2 (12 + 1964)
= 977/2 × 1976
= 977 × 1976/2
= 1930552/2 = 965276
Thus, the sum of all terms of the given even numbers from 12 to 1964 = 965276
And, the total number of terms = 977
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 1964
= 965276/977 = 988
Thus, the average of the given even numbers from 12 to 1964 = 988 Answer
Similar Questions
(1) Find the average of the first 1792 odd numbers.
(2) Find the average of even numbers from 10 to 314
(3) Find the average of even numbers from 6 to 162
(4) What will be the average of the first 4246 odd numbers?
(5) What will be the average of the first 4163 odd numbers?
(6) Find the average of odd numbers from 5 to 61
(7) Find the average of odd numbers from 15 to 243
(8) Find the average of odd numbers from 11 to 1345
(9) Find the average of odd numbers from 9 to 313
(10) Find the average of odd numbers from 9 to 1063