Question:
Find the average of even numbers from 12 to 1364
Correct Answer
688
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1364
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1364 are
12, 14, 16, . . . . 1364
After observing the above list of the even numbers from 12 to 1364 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1364 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1364
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1364
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 1364
= 12 + 1364/2
= 1376/2 = 688
Thus, the average of the even numbers from 12 to 1364 = 688 Answer
Method (2) to find the average of the even numbers from 12 to 1364
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1364 are
12, 14, 16, . . . . 1364
The even numbers from 12 to 1364 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1364
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 1364
1364 = 12 + (n – 1) × 2
⇒ 1364 = 12 + 2 n – 2
⇒ 1364 = 12 – 2 + 2 n
⇒ 1364 = 10 + 2 n
After transposing 10 to LHS
⇒ 1364 – 10 = 2 n
⇒ 1354 = 2 n
After rearranging the above expression
⇒ 2 n = 1354
After transposing 2 to RHS
⇒ n = 1354/2
⇒ n = 677
Thus, the number of terms of even numbers from 12 to 1364 = 677
This means 1364 is the 677th term.
Finding the sum of the given even numbers from 12 to 1364
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 1364
= 677/2 (12 + 1364)
= 677/2 × 1376
= 677 × 1376/2
= 931552/2 = 465776
Thus, the sum of all terms of the given even numbers from 12 to 1364 = 465776
And, the total number of terms = 677
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 1364
= 465776/677 = 688
Thus, the average of the given even numbers from 12 to 1364 = 688 Answer
Similar Questions
(1) What will be the average of the first 4201 odd numbers?
(2) Find the average of odd numbers from 13 to 467
(3) Find the average of the first 2579 odd numbers.
(4) Find the average of even numbers from 6 to 1436
(5) Find the average of even numbers from 6 to 1874
(6) Find the average of odd numbers from 3 to 1421
(7) Find the average of the first 3693 even numbers.
(8) Find the average of the first 4213 even numbers.
(9) Find the average of the first 2488 even numbers.
(10) Find the average of odd numbers from 11 to 1055