Question : Find the average of even numbers from 12 to 268
Correct Answer 140
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 268
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 268 are
12, 14, 16, . . . . 268
After observing the above list of the even numbers from 12 to 268 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 268 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 268
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 268
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 268
= 12 + 268/2
= 280/2 = 140
Thus, the average of the even numbers from 12 to 268 = 140 Answer
Method (2) to find the average of the even numbers from 12 to 268
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 268 are
12, 14, 16, . . . . 268
The even numbers from 12 to 268 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 268
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 268
268 = 12 + (n – 1) × 2
⇒ 268 = 12 + 2 n – 2
⇒ 268 = 12 – 2 + 2 n
⇒ 268 = 10 + 2 n
After transposing 10 to LHS
⇒ 268 – 10 = 2 n
⇒ 258 = 2 n
After rearranging the above expression
⇒ 2 n = 258
After transposing 2 to RHS
⇒ n = 258/2
⇒ n = 129
Thus, the number of terms of even numbers from 12 to 268 = 129
This means 268 is the 129th term.
Finding the sum of the given even numbers from 12 to 268
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 268
= 129/2 (12 + 268)
= 129/2 × 280
= 129 × 280/2
= 36120/2 = 18060
Thus, the sum of all terms of the given even numbers from 12 to 268 = 18060
And, the total number of terms = 129
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 268
= 18060/129 = 140
Thus, the average of the given even numbers from 12 to 268 = 140 Answer
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