Question : Find the average of even numbers from 8 to 1412
Correct Answer 710
Solution & Explanation
Solution
Method (1) to find the average of the even numbers from 8 to 1412
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 8 to 1412 are
8, 10, 12, . . . . 1412
After observing the above list of the even numbers from 8 to 1412 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 8 to 1412 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 8 to 1412
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1412
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 8 to 1412
= 8 + 1412/2
= 1420/2 = 710
Thus, the average of the even numbers from 8 to 1412 = 710 Answer
Method (2) to find the average of the even numbers from 8 to 1412
Finding the average of given continuous even numbers after finding their sum
The even numbers from 8 to 1412 are
8, 10, 12, . . . . 1412
The even numbers from 8 to 1412 form an Arithmetic Series in which
The First Term (a) = 8
The Common Difference (d) = 2
And the last term (ℓ) = 1412
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 8 to 1412
1412 = 8 + (n – 1) × 2
⇒ 1412 = 8 + 2 n – 2
⇒ 1412 = 8 – 2 + 2 n
⇒ 1412 = 6 + 2 n
After transposing 6 to LHS
⇒ 1412 – 6 = 2 n
⇒ 1406 = 2 n
After rearranging the above expression
⇒ 2 n = 1406
After transposing 2 to RHS
⇒ n = 1406/2
⇒ n = 703
Thus, the number of terms of even numbers from 8 to 1412 = 703
This means 1412 is the 703th term.
Finding the sum of the given even numbers from 8 to 1412
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 8 to 1412
= 703/2 (8 + 1412)
= 703/2 × 1420
= 703 × 1420/2
= 998260/2 = 499130
Thus, the sum of all terms of the given even numbers from 8 to 1412 = 499130
And, the total number of terms = 703
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 8 to 1412
= 499130/703 = 710
Thus, the average of the given even numbers from 8 to 1412 = 710 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 924
(2) Find the average of odd numbers from 13 to 875
(3) Find the average of odd numbers from 11 to 1201
(4) What is the average of the first 1169 even numbers?
(5) Find the average of odd numbers from 13 to 467
(6) Find the average of the first 3053 odd numbers.
(7) What is the average of the first 1214 even numbers?
(8) Find the average of even numbers from 12 to 1428
(9) Find the average of the first 3128 odd numbers.
(10) Find the average of the first 2032 even numbers.