Question:
Find the average of even numbers from 10 to 1608
Correct Answer
809
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 10 to 1608
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 10 to 1608 are
10, 12, 14, . . . . 1608
After observing the above list of the even numbers from 10 to 1608 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 10 to 1608 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 10 to 1608
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1608
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 1608
= 10 + 1608/2
= 1618/2 = 809
Thus, the average of the even numbers from 10 to 1608 = 809 Answer
Method (2) to find the average of the even numbers from 10 to 1608
Finding the average of given continuous even numbers after finding their sum
The even numbers from 10 to 1608 are
10, 12, 14, . . . . 1608
The even numbers from 10 to 1608 form an Arithmetic Series in which
The First Term (a) = 10
The Common Difference (d) = 2
And the last term (ℓ) = 1608
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 1608
1608 = 10 + (n – 1) × 2
⇒ 1608 = 10 + 2 n – 2
⇒ 1608 = 10 – 2 + 2 n
⇒ 1608 = 8 + 2 n
After transposing 8 to LHS
⇒ 1608 – 8 = 2 n
⇒ 1600 = 2 n
After rearranging the above expression
⇒ 2 n = 1600
After transposing 2 to RHS
⇒ n = 1600/2
⇒ n = 800
Thus, the number of terms of even numbers from 10 to 1608 = 800
This means 1608 is the 800th term.
Finding the sum of the given even numbers from 10 to 1608
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 1608
= 800/2 (10 + 1608)
= 800/2 × 1618
= 800 × 1618/2
= 1294400/2 = 647200
Thus, the sum of all terms of the given even numbers from 10 to 1608 = 647200
And, the total number of terms = 800
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 1608
= 647200/800 = 809
Thus, the average of the given even numbers from 10 to 1608 = 809 Answer
Similar Questions
(1) What will be the average of the first 4092 odd numbers?
(2) Find the average of even numbers from 4 to 1444
(3) What is the average of the first 235 even numbers?
(4) What is the average of the first 566 even numbers?
(5) Find the average of even numbers from 12 to 1650
(6) What is the average of the first 1453 even numbers?
(7) Find the average of the first 3406 odd numbers.
(8) What will be the average of the first 4830 odd numbers?
(9) Find the average of the first 2999 odd numbers.
(10) Find the average of the first 2440 odd numbers.