Question:
Find the average of even numbers from 10 to 298
Correct Answer
154
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 298
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 298 are
10, 12, 14, . . . . 298
After observing the above list of the even numbers from 10 to 298 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 298 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 298
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 298
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 298
= 10 + 298/2
= 308/2 = 154
Thus, the average of the even numbers from 10 to 298 = 154 Answer
Method (2) to find the average of the even numbers from 10 to 298
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 298 are
10, 12, 14, . . . . 298
The even numbers from 10 to 298 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 298
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 298
298 = 10 + (n – 1) × 2
⇒ 298 = 10 + 2 n – 2
⇒ 298 = 10 – 2 + 2 n
⇒ 298 = 8 + 2 n
After transposing 8 to LHS
⇒ 298 – 8 = 2 n
⇒ 290 = 2 n
After rearranging the above expression
⇒ 2 n = 290
After transposing 2 to RHS
⇒ n = 290/2
⇒ n = 145
Thus, the number of terms of even numbers from 10 to 298 = 145
This means 298 is the 145th term.
Finding the sum of the given even numbers from 10 to 298
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 298
= 145/2 (10 + 298)
= 145/2 × 308
= 145 × 308/2
= 44660/2 = 22330
Thus, the sum of all terms of the given even numbers from 10 to 298 = 22330
And, the total number of terms = 145
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 298
= 22330/145 = 154
Thus, the average of the given even numbers from 10 to 298 = 154 Answer
Similar Questions
(1) Find the average of odd numbers from 3 to 77
(2) Find the average of even numbers from 6 to 480
(3) Find the average of the first 3343 even numbers.
(4) Find the average of odd numbers from 15 to 1743
(5) Find the average of the first 2997 even numbers.
(6) Find the average of even numbers from 4 to 896
(7) What will be the average of the first 4256 odd numbers?
(8) Find the average of the first 4345 even numbers.
(9) Find the average of the first 2131 even numbers.
(10) Find the average of even numbers from 12 to 1050