Question:
Find the average of even numbers from 10 to 1324
Correct Answer
667
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1324
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1324 are
10, 12, 14, . . . . 1324
After observing the above list of the even numbers from 10 to 1324 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1324 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1324
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1324
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 10 to 1324
= 10 + 1324/2
= 1334/2 = 667
Thus, the average of the even numbers from 10 to 1324 = 667 Answer
Method (2) to find the average of the even numbers from 10 to 1324
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1324 are
10, 12, 14, . . . . 1324
The even numbers from 10 to 1324 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1324
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 10 to 1324
1324 = 10 + (n – 1) × 2
⇒ 1324 = 10 + 2 n – 2
⇒ 1324 = 10 – 2 + 2 n
⇒ 1324 = 8 + 2 n
After transposing 8 to LHS
⇒ 1324 – 8 = 2 n
⇒ 1316 = 2 n
After rearranging the above expression
⇒ 2 n = 1316
After transposing 2 to RHS
⇒ n = 1316/2
⇒ n = 658
Thus, the number of terms of even numbers from 10 to 1324 = 658
This means 1324 is the 658th term.
Finding the sum of the given even numbers from 10 to 1324
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 10 to 1324
= 658/2 (10 + 1324)
= 658/2 × 1334
= 658 × 1334/2
= 877772/2 = 438886
Thus, the sum of all terms of the given even numbers from 10 to 1324 = 438886
And, the total number of terms = 658
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 10 to 1324
= 438886/658 = 667
Thus, the average of the given even numbers from 10 to 1324 = 667 Answer
Similar Questions
(1) What is the average of the first 734 even numbers?
(2) Find the average of odd numbers from 9 to 1151
(3) What is the average of the first 523 even numbers?
(4) Find the average of the first 3891 even numbers.
(5) What is the average of the first 1932 even numbers?
(6) What is the average of the first 1408 even numbers?
(7) What is the average of the first 684 even numbers?
(8) Find the average of even numbers from 4 to 1092
(9) Find the average of the first 2820 even numbers.
(10) Find the average of odd numbers from 11 to 267