Question : Find the average of even numbers from 12 to 1112
Correct Answer 562
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1112
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1112 are
12, 14, 16, . . . . 1112
After observing the above list of the even numbers from 12 to 1112 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1112 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1112
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1112
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 1112
= 12 + 1112/2
= 1124/2 = 562
Thus, the average of the even numbers from 12 to 1112 = 562 Answer
Method (2) to find the average of the even numbers from 12 to 1112
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1112 are
12, 14, 16, . . . . 1112
The even numbers from 12 to 1112 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1112
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 1112
1112 = 12 + (n – 1) × 2
⇒ 1112 = 12 + 2 n – 2
⇒ 1112 = 12 – 2 + 2 n
⇒ 1112 = 10 + 2 n
After transposing 10 to LHS
⇒ 1112 – 10 = 2 n
⇒ 1102 = 2 n
After rearranging the above expression
⇒ 2 n = 1102
After transposing 2 to RHS
⇒ n = 1102/2
⇒ n = 551
Thus, the number of terms of even numbers from 12 to 1112 = 551
This means 1112 is the 551th term.
Finding the sum of the given even numbers from 12 to 1112
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 1112
= 551/2 (12 + 1112)
= 551/2 × 1124
= 551 × 1124/2
= 619324/2 = 309662
Thus, the sum of all terms of the given even numbers from 12 to 1112 = 309662
And, the total number of terms = 551
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 1112
= 309662/551 = 562
Thus, the average of the given even numbers from 12 to 1112 = 562 Answer
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