Question : Find the average of even numbers from 8 to 910
Correct Answer 459
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 910
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 910 are
8, 10, 12, . . . . 910
After observing the above list of the even numbers from 8 to 910 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 910 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 910
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 910
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 8 to 910
= 8 + 910/2
= 918/2 = 459
Thus, the average of the even numbers from 8 to 910 = 459 Answer
Method (2) to find the average of the even numbers from 8 to 910
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 910 are
8, 10, 12, . . . . 910
The even numbers from 8 to 910 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 910
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 8 to 910
910 = 8 + (n – 1) × 2
⇒ 910 = 8 + 2 n – 2
⇒ 910 = 8 – 2 + 2 n
⇒ 910 = 6 + 2 n
After transposing 6 to LHS
⇒ 910 – 6 = 2 n
⇒ 904 = 2 n
After rearranging the above expression
⇒ 2 n = 904
After transposing 2 to RHS
⇒ n = 904/2
⇒ n = 452
Thus, the number of terms of even numbers from 8 to 910 = 452
This means 910 is the 452th term.
Finding the sum of the given even numbers from 8 to 910
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 8 to 910
= 452/2 (8 + 910)
= 452/2 × 918
= 452 × 918/2
= 414936/2 = 207468
Thus, the sum of all terms of the given even numbers from 8 to 910 = 207468
And, the total number of terms = 452
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 8 to 910
= 207468/452 = 459
Thus, the average of the given even numbers from 8 to 910 = 459 Answer
Similar Questions
(1) Find the average of even numbers from 12 to 142
(2) Find the average of the first 816 odd numbers.
(3) Find the average of odd numbers from 5 to 379
(4) What will be the average of the first 4143 odd numbers?
(5) Find the average of odd numbers from 9 to 69
(6) Find the average of the first 503 odd numbers.
(7) What will be the average of the first 4050 odd numbers?
(8) Find the average of odd numbers from 9 to 1321
(9) Find the average of the first 2353 odd numbers.
(10) Find the average of the first 2628 even numbers.