Question:
Find the average of even numbers from 12 to 1246
Correct Answer
629
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1246
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1246 are
12, 14, 16, . . . . 1246
After observing the above list of the even numbers from 12 to 1246 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1246 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1246
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1246
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 1246
= 12 + 1246/2
= 1258/2 = 629
Thus, the average of the even numbers from 12 to 1246 = 629 Answer
Method (2) to find the average of the even numbers from 12 to 1246
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1246 are
12, 14, 16, . . . . 1246
The even numbers from 12 to 1246 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1246
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 1246
1246 = 12 + (n – 1) × 2
⇒ 1246 = 12 + 2 n – 2
⇒ 1246 = 12 – 2 + 2 n
⇒ 1246 = 10 + 2 n
After transposing 10 to LHS
⇒ 1246 – 10 = 2 n
⇒ 1236 = 2 n
After rearranging the above expression
⇒ 2 n = 1236
After transposing 2 to RHS
⇒ n = 1236/2
⇒ n = 618
Thus, the number of terms of even numbers from 12 to 1246 = 618
This means 1246 is the 618th term.
Finding the sum of the given even numbers from 12 to 1246
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 1246
= 618/2 (12 + 1246)
= 618/2 × 1258
= 618 × 1258/2
= 777444/2 = 388722
Thus, the sum of all terms of the given even numbers from 12 to 1246 = 388722
And, the total number of terms = 618
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 1246
= 388722/618 = 629
Thus, the average of the given even numbers from 12 to 1246 = 629 Answer
Similar Questions
(1) What will be the average of the first 4188 odd numbers?
(2) Find the average of the first 1906 odd numbers.
(3) Find the average of odd numbers from 15 to 1007
(4) Find the average of the first 3147 odd numbers.
(5) Find the average of even numbers from 12 to 792
(6) Find the average of the first 2414 even numbers.
(7) What is the average of the first 84 even numbers?
(8) Find the average of the first 2454 odd numbers.
(9) Find the average of odd numbers from 11 to 781
(10) Find the average of odd numbers from 5 to 825