Question : Find the average of even numbers from 10 to 956
Correct Answer 483
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 956
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 956 are
10, 12, 14, . . . . 956
After observing the above list of the even numbers from 10 to 956 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 956 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 956
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 956
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 10 to 956
= 10 + 956/2
= 966/2 = 483
Thus, the average of the even numbers from 10 to 956 = 483 Answer
Method (2) to find the average of the even numbers from 10 to 956
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 956 are
10, 12, 14, . . . . 956
The even numbers from 10 to 956 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 956
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 10 to 956
956 = 10 + (n – 1) × 2
⇒ 956 = 10 + 2 n – 2
⇒ 956 = 10 – 2 + 2 n
⇒ 956 = 8 + 2 n
After transposing 8 to LHS
⇒ 956 – 8 = 2 n
⇒ 948 = 2 n
After rearranging the above expression
⇒ 2 n = 948
After transposing 2 to RHS
⇒ n = 948/2
⇒ n = 474
Thus, the number of terms of even numbers from 10 to 956 = 474
This means 956 is the 474th term.
Finding the sum of the given even numbers from 10 to 956
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 10 to 956
= 474/2 (10 + 956)
= 474/2 × 966
= 474 × 966/2
= 457884/2 = 228942
Thus, the sum of all terms of the given even numbers from 10 to 956 = 228942
And, the total number of terms = 474
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 10 to 956
= 228942/474 = 483
Thus, the average of the given even numbers from 10 to 956 = 483 Answer
Similar Questions
(1) Find the average of the first 4052 even numbers.
(2) Find the average of the first 3394 odd numbers.
(3) What will be the average of the first 4406 odd numbers?
(4) What is the average of the first 1092 even numbers?
(5) Find the average of even numbers from 6 to 1076
(6) Find the average of the first 4247 even numbers.
(7) What will be the average of the first 4310 odd numbers?
(8) Find the average of the first 1614 odd numbers.
(9) Find the average of the first 3996 even numbers.
(10) Find the average of the first 2983 odd numbers.