Question:
Find the average of even numbers from 10 to 552
Correct Answer
281
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 552
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 552 are
10, 12, 14, . . . . 552
After observing the above list of the even numbers from 10 to 552 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 552 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 552
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 552
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 552
= 10 + 552/2
= 562/2 = 281
Thus, the average of the even numbers from 10 to 552 = 281 Answer
Method (2) to find the average of the even numbers from 10 to 552
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 552 are
10, 12, 14, . . . . 552
The even numbers from 10 to 552 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 552
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 552
552 = 10 + (n – 1) × 2
⇒ 552 = 10 + 2 n – 2
⇒ 552 = 10 – 2 + 2 n
⇒ 552 = 8 + 2 n
After transposing 8 to LHS
⇒ 552 – 8 = 2 n
⇒ 544 = 2 n
After rearranging the above expression
⇒ 2 n = 544
After transposing 2 to RHS
⇒ n = 544/2
⇒ n = 272
Thus, the number of terms of even numbers from 10 to 552 = 272
This means 552 is the 272th term.
Finding the sum of the given even numbers from 10 to 552
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 552
= 272/2 (10 + 552)
= 272/2 × 562
= 272 × 562/2
= 152864/2 = 76432
Thus, the sum of all terms of the given even numbers from 10 to 552 = 76432
And, the total number of terms = 272
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 552
= 76432/272 = 281
Thus, the average of the given even numbers from 10 to 552 = 281 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 589
(2) Find the average of the first 1396 odd numbers.
(3) Find the average of odd numbers from 5 to 343
(4) Find the average of the first 1563 odd numbers.
(5) Find the average of odd numbers from 5 to 1051
(6) Find the average of odd numbers from 5 to 1083
(7) Find the average of odd numbers from 13 to 285
(8) Find the average of odd numbers from 13 to 1495
(9) What will be the average of the first 4026 odd numbers?
(10) Find the average of the first 2526 odd numbers.