Question : Find the average of even numbers from 10 to 978
Correct Answer 494
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 978
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 978 are
10, 12, 14, . . . . 978
After observing the above list of the even numbers from 10 to 978 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 978 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 978
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 978
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 978
= 10 + 978/2
= 988/2 = 494
Thus, the average of the even numbers from 10 to 978 = 494 Answer
Method (2) to find the average of the even numbers from 10 to 978
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 978 are
10, 12, 14, . . . . 978
The even numbers from 10 to 978 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 978
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 978
978 = 10 + (n – 1) × 2
⇒ 978 = 10 + 2 n – 2
⇒ 978 = 10 – 2 + 2 n
⇒ 978 = 8 + 2 n
After transposing 8 to LHS
⇒ 978 – 8 = 2 n
⇒ 970 = 2 n
After rearranging the above expression
⇒ 2 n = 970
After transposing 2 to RHS
⇒ n = 970/2
⇒ n = 485
Thus, the number of terms of even numbers from 10 to 978 = 485
This means 978 is the 485th term.
Finding the sum of the given even numbers from 10 to 978
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 978
= 485/2 (10 + 978)
= 485/2 × 988
= 485 × 988/2
= 479180/2 = 239590
Thus, the sum of all terms of the given even numbers from 10 to 978 = 239590
And, the total number of terms = 485
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 978
= 239590/485 = 494
Thus, the average of the given even numbers from 10 to 978 = 494 Answer
Similar Questions
(1) What is the average of the first 58 even numbers?
(2) Find the average of odd numbers from 11 to 515
(3) Find the average of the first 4812 even numbers.
(4) Find the average of the first 3024 even numbers.
(5) What will be the average of the first 4999 odd numbers?
(6) Find the average of the first 3920 odd numbers.
(7) Find the average of odd numbers from 11 to 887
(8) Find the average of even numbers from 10 to 356
(9) Find the average of the first 722 odd numbers.
(10) Find the average of the first 3869 even numbers.