Question:
Find the average of even numbers from 12 to 1550
Correct Answer
781
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1550
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1550 are
12, 14, 16, . . . . 1550
After observing the above list of the even numbers from 12 to 1550 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1550 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1550
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1550
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 1550
= 12 + 1550/2
= 1562/2 = 781
Thus, the average of the even numbers from 12 to 1550 = 781 Answer
Method (2) to find the average of the even numbers from 12 to 1550
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1550 are
12, 14, 16, . . . . 1550
The even numbers from 12 to 1550 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1550
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 1550
1550 = 12 + (n – 1) × 2
⇒ 1550 = 12 + 2 n – 2
⇒ 1550 = 12 – 2 + 2 n
⇒ 1550 = 10 + 2 n
After transposing 10 to LHS
⇒ 1550 – 10 = 2 n
⇒ 1540 = 2 n
After rearranging the above expression
⇒ 2 n = 1540
After transposing 2 to RHS
⇒ n = 1540/2
⇒ n = 770
Thus, the number of terms of even numbers from 12 to 1550 = 770
This means 1550 is the 770th term.
Finding the sum of the given even numbers from 12 to 1550
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 1550
= 770/2 (12 + 1550)
= 770/2 × 1562
= 770 × 1562/2
= 1202740/2 = 601370
Thus, the sum of all terms of the given even numbers from 12 to 1550 = 601370
And, the total number of terms = 770
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 1550
= 601370/770 = 781
Thus, the average of the given even numbers from 12 to 1550 = 781 Answer
Similar Questions
(1) Find the average of odd numbers from 15 to 1613
(2) What will be the average of the first 4717 odd numbers?
(3) Find the average of the first 2078 odd numbers.
(4) Find the average of the first 241 odd numbers.
(5) Find the average of the first 490 odd numbers.
(6) Find the average of odd numbers from 15 to 109
(7) Find the average of even numbers from 8 to 64
(8) Find the average of odd numbers from 3 to 809
(9) What is the average of the first 1488 even numbers?
(10) Find the average of even numbers from 8 to 282