Question:
Find the average of even numbers from 10 to 1406
Correct Answer
708
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1406
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1406 are
10, 12, 14, . . . . 1406
After observing the above list of the even numbers from 10 to 1406 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1406 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1406
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1406
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 1406
= 10 + 1406/2
= 1416/2 = 708
Thus, the average of the even numbers from 10 to 1406 = 708 Answer
Method (2) to find the average of the even numbers from 10 to 1406
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1406 are
10, 12, 14, . . . . 1406
The even numbers from 10 to 1406 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1406
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 1406
1406 = 10 + (n – 1) × 2
⇒ 1406 = 10 + 2 n – 2
⇒ 1406 = 10 – 2 + 2 n
⇒ 1406 = 8 + 2 n
After transposing 8 to LHS
⇒ 1406 – 8 = 2 n
⇒ 1398 = 2 n
After rearranging the above expression
⇒ 2 n = 1398
After transposing 2 to RHS
⇒ n = 1398/2
⇒ n = 699
Thus, the number of terms of even numbers from 10 to 1406 = 699
This means 1406 is the 699th term.
Finding the sum of the given even numbers from 10 to 1406
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 1406
= 699/2 (10 + 1406)
= 699/2 × 1416
= 699 × 1416/2
= 989784/2 = 494892
Thus, the sum of all terms of the given even numbers from 10 to 1406 = 494892
And, the total number of terms = 699
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 1406
= 494892/699 = 708
Thus, the average of the given even numbers from 10 to 1406 = 708 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 273
(2) Find the average of the first 1887 odd numbers.
(3) Find the average of odd numbers from 15 to 99
(4) Find the average of the first 4400 even numbers.
(5) Find the average of even numbers from 10 to 762
(6) Find the average of the first 1404 odd numbers.
(7) Find the average of the first 4085 even numbers.
(8) What will be the average of the first 4608 odd numbers?
(9) Find the average of the first 3287 even numbers.
(10) Find the average of even numbers from 6 to 960