Question:
Find the average of even numbers from 10 to 234
Correct Answer
122
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 234
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 234 are
10, 12, 14, . . . . 234
After observing the above list of the even numbers from 10 to 234 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 234 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 234
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 234
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 234
= 10 + 234/2
= 244/2 = 122
Thus, the average of the even numbers from 10 to 234 = 122 Answer
Method (2) to find the average of the even numbers from 10 to 234
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 234 are
10, 12, 14, . . . . 234
The even numbers from 10 to 234 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 234
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 234
234 = 10 + (n – 1) × 2
⇒ 234 = 10 + 2 n – 2
⇒ 234 = 10 – 2 + 2 n
⇒ 234 = 8 + 2 n
After transposing 8 to LHS
⇒ 234 – 8 = 2 n
⇒ 226 = 2 n
After rearranging the above expression
⇒ 2 n = 226
After transposing 2 to RHS
⇒ n = 226/2
⇒ n = 113
Thus, the number of terms of even numbers from 10 to 234 = 113
This means 234 is the 113th term.
Finding the sum of the given even numbers from 10 to 234
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 234
= 113/2 (10 + 234)
= 113/2 × 244
= 113 × 244/2
= 27572/2 = 13786
Thus, the sum of all terms of the given even numbers from 10 to 234 = 13786
And, the total number of terms = 113
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 234
= 13786/113 = 122
Thus, the average of the given even numbers from 10 to 234 = 122 Answer
Similar Questions
(1) Find the average of odd numbers from 3 to 239
(2) What is the average of the first 20 even numbers?
(3) Find the average of even numbers from 12 to 1504
(4) Find the average of the first 2517 odd numbers.
(5) Find the average of odd numbers from 11 to 1421
(6) Find the average of even numbers from 10 to 1594
(7) Find the average of odd numbers from 13 to 577
(8) Find the average of even numbers from 10 to 812
(9) What is the average of the first 1746 even numbers?
(10) Find the average of even numbers from 6 to 1768