Question:
Find the average of even numbers from 12 to 234
Correct Answer
123
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 234
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 234 are
12, 14, 16, . . . . 234
After observing the above list of the even numbers from 12 to 234 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 234 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 234
The First Term (a) = 12
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 12 to 234
= 12 + 234/2
= 246/2 = 123
Thus, the average of the even numbers from 12 to 234 = 123 Answer
Method (2) to find the average of the even numbers from 12 to 234
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 234 are
12, 14, 16, . . . . 234
The even numbers from 12 to 234 form an Arithmetic Series in which
The First Term (a) = 12
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 12 to 234
234 = 12 + (n – 1) × 2
⇒ 234 = 12 + 2 n – 2
⇒ 234 = 12 – 2 + 2 n
⇒ 234 = 10 + 2 n
After transposing 10 to LHS
⇒ 234 – 10 = 2 n
⇒ 224 = 2 n
After rearranging the above expression
⇒ 2 n = 224
After transposing 2 to RHS
⇒ n = 224/2
⇒ n = 112
Thus, the number of terms of even numbers from 12 to 234 = 112
This means 234 is the 112th term.
Finding the sum of the given even numbers from 12 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 12 to 234
= 112/2 (12 + 234)
= 112/2 × 246
= 112 × 246/2
= 27552/2 = 13776
Thus, the sum of all terms of the given even numbers from 12 to 234 = 13776
And, the total number of terms = 112
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 12 to 234
= 13776/112 = 123
Thus, the average of the given even numbers from 12 to 234 = 123 Answer
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