Question:
Find the average of even numbers from 10 to 482
Correct Answer
246
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 482
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 482 are
10, 12, 14, . . . . 482
After observing the above list of the even numbers from 10 to 482 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 482 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 482
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 482
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 482
= 10 + 482/2
= 492/2 = 246
Thus, the average of the even numbers from 10 to 482 = 246 Answer
Method (2) to find the average of the even numbers from 10 to 482
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 482 are
10, 12, 14, . . . . 482
The even numbers from 10 to 482 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 482
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 482
482 = 10 + (n – 1) × 2
⇒ 482 = 10 + 2 n – 2
⇒ 482 = 10 – 2 + 2 n
⇒ 482 = 8 + 2 n
After transposing 8 to LHS
⇒ 482 – 8 = 2 n
⇒ 474 = 2 n
After rearranging the above expression
⇒ 2 n = 474
After transposing 2 to RHS
⇒ n = 474/2
⇒ n = 237
Thus, the number of terms of even numbers from 10 to 482 = 237
This means 482 is the 237th term.
Finding the sum of the given even numbers from 10 to 482
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 482
= 237/2 (10 + 482)
= 237/2 × 492
= 237 × 492/2
= 116604/2 = 58302
Thus, the sum of all terms of the given even numbers from 10 to 482 = 58302
And, the total number of terms = 237
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 482
= 58302/237 = 246
Thus, the average of the given even numbers from 10 to 482 = 246 Answer
Similar Questions
(1) What will be the average of the first 4545 odd numbers?
(2) Find the average of the first 1269 odd numbers.
(3) Find the average of even numbers from 10 to 256
(4) Find the average of the first 3192 even numbers.
(5) Find the average of the first 3838 even numbers.
(6) Find the average of odd numbers from 5 to 1191
(7) What will be the average of the first 4571 odd numbers?
(8) Find the average of the first 3745 even numbers.
(9) Find the average of the first 2158 even numbers.
(10) What will be the average of the first 4973 odd numbers?