Question:
Find the average of even numbers from 12 to 1670
Correct Answer
841
Solution And Explanation
Solution
Method (1) to find the average of the even numbers from 12 to 1670
Shortcut Trick to find the average of the given continuous even numbers
The even numbers from 12 to 1670 are
12, 14, 16, . . . . 1670
After observing the above list of the even numbers from 12 to 1670 we find that the difference between two consecutive terms are equal. This means the list of the even numbers from 12 to 1670 form an Arithmetic Series.
In the Arithmetic Series of the even numbers from 12 to 1670
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1670
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 1670
= 12 + 1670/2
= 1682/2 = 841
Thus, the average of the even numbers from 12 to 1670 = 841 Answer
Method (2) to find the average of the even numbers from 12 to 1670
Finding the average of given continuous even numbers after finding their sum
The even numbers from 12 to 1670 are
12, 14, 16, . . . . 1670
The even numbers from 12 to 1670 form an Arithmetic Series in which
The First Term (a) = 12
The Common Difference (d) = 2
And the last term (ℓ) = 1670
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 1670
1670 = 12 + (n – 1) × 2
⇒ 1670 = 12 + 2 n – 2
⇒ 1670 = 12 – 2 + 2 n
⇒ 1670 = 10 + 2 n
After transposing 10 to LHS
⇒ 1670 – 10 = 2 n
⇒ 1660 = 2 n
After rearranging the above expression
⇒ 2 n = 1660
After transposing 2 to RHS
⇒ n = 1660/2
⇒ n = 830
Thus, the number of terms of even numbers from 12 to 1670 = 830
This means 1670 is the 830th term.
Finding the sum of the given even numbers from 12 to 1670
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 1670
= 830/2 (12 + 1670)
= 830/2 × 1682
= 830 × 1682/2
= 1396060/2 = 698030
Thus, the sum of all terms of the given even numbers from 12 to 1670 = 698030
And, the total number of terms = 830
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 1670
= 698030/830 = 841
Thus, the average of the given even numbers from 12 to 1670 = 841 Answer
Similar Questions
(1) Find the average of the first 238 odd numbers.
(2) Find the average of odd numbers from 3 to 1207
(3) Find the average of the first 4369 even numbers.
(4) Find the average of odd numbers from 7 to 1019
(5) Find the average of the first 3074 even numbers.
(6) Find the average of odd numbers from 5 to 145
(7) Find the average of even numbers from 6 to 638
(8) Find the average of the first 3902 even numbers.
(9) Find the average of the first 1588 odd numbers.
(10) Find the average of even numbers from 8 to 1146