Question:
Find the average of even numbers from 6 to 970
Correct Answer
488
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 6 to 970
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 6 to 970 are
6, 8, 10, . . . . 970
After observing the above list of the even numbers from 6 to 970 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 6 to 970 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 6 to 970
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 970
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 6 to 970
= 6 + 970/2
= 976/2 = 488
Thus, the average of the even numbers from 6 to 970 = 488 Answer
Method (2) to find the average of the even numbers from 6 to 970
Finding the average of given continuous even numbers after finding their sum
The even numbers from 6 to 970 are
6, 8, 10, . . . . 970
The even numbers from 6 to 970 form an Arithmetic Series in which
The First Term (a) = 6
The Common Difference (d) = 2
And the last term (ℓ) = 970
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 6 to 970
970 = 6 + (n – 1) × 2
⇒ 970 = 6 + 2 n – 2
⇒ 970 = 6 – 2 + 2 n
⇒ 970 = 4 + 2 n
After transposing 4 to LHS
⇒ 970 – 4 = 2 n
⇒ 966 = 2 n
After rearranging the above expression
⇒ 2 n = 966
After transposing 2 to RHS
⇒ n = 966/2
⇒ n = 483
Thus, the number of terms of even numbers from 6 to 970 = 483
This means 970 is the 483th term.
Finding the sum of the given even numbers from 6 to 970
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 6 to 970
= 483/2 (6 + 970)
= 483/2 × 976
= 483 × 976/2
= 471408/2 = 235704
Thus, the sum of all terms of the given even numbers from 6 to 970 = 235704
And, the total number of terms = 483
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 6 to 970
= 235704/483 = 488
Thus, the average of the given even numbers from 6 to 970 = 488 Answer
Similar Questions
(1) Find the average of the first 1584 odd numbers.
(2) Find the average of even numbers from 6 to 722
(3) Find the average of odd numbers from 13 to 1027
(4) What is the average of the first 528 even numbers?
(5) Find the average of odd numbers from 13 to 71
(6) Find the average of the first 3239 odd numbers.
(7) Find the average of odd numbers from 13 to 205
(8) Find the average of even numbers from 6 to 820
(9) What is the average of the first 1223 even numbers?
(10) Find the average of the first 2296 even numbers.