Question : Find the average of even numbers from 8 to 250
Correct Answer 129
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 250
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 250 are
8, 10, 12, . . . . 250
After observing the above list of the even numbers from 8 to 250 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 250 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 250
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 250
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 250
= 8 + 250/2
= 258/2 = 129
Thus, the average of the even numbers from 8 to 250 = 129 Answer
Method (2) to find the average of the even numbers from 8 to 250
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 250 are
8, 10, 12, . . . . 250
The even numbers from 8 to 250 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 250
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 250
250 = 8 + (n – 1) × 2
⇒ 250 = 8 + 2 n – 2
⇒ 250 = 8 – 2 + 2 n
⇒ 250 = 6 + 2 n
After transposing 6 to LHS
⇒ 250 – 6 = 2 n
⇒ 244 = 2 n
After rearranging the above expression
⇒ 2 n = 244
After transposing 2 to RHS
⇒ n = 244/2
⇒ n = 122
Thus, the number of terms of even numbers from 8 to 250 = 122
This means 250 is the 122th term.
Finding the sum of the given even numbers from 8 to 250
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 250
= 122/2 (8 + 250)
= 122/2 × 258
= 122 × 258/2
= 31476/2 = 15738
Thus, the sum of all terms of the given even numbers from 8 to 250 = 15738
And, the total number of terms = 122
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 250
= 15738/122 = 129
Thus, the average of the given even numbers from 8 to 250 = 129 Answer
Similar Questions
(1) Find the average of the first 1547 odd numbers.
(2) Find the average of odd numbers from 7 to 183
(3) Find the average of the first 381 odd numbers.
(4) Find the average of the first 2609 odd numbers.
(5) Find the average of even numbers from 12 to 1216
(6) What is the average of the first 1953 even numbers?
(7) Find the average of even numbers from 8 to 1376
(8) Find the average of the first 214 odd numbers.
(9) Find the average of the first 3587 odd numbers.
(10) Find the average of odd numbers from 9 to 355