Question:
Find the average of odd numbers from 15 to 1667
Correct Answer
841
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1667
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1667 are
15, 17, 19, . . . . 1667
After observing the above list of the odd numbers from 15 to 1667 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1667 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1667
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1667
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 odd numbers from 15 to 1667
= 15 + 1667/2
= 1682/2 = 841
Thus, the average of the odd numbers from 15 to 1667 = 841 Answer
Method (2) to find the average of the odd numbers from 15 to 1667
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1667 are
15, 17, 19, . . . . 1667
The odd numbers from 15 to 1667 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1667
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 odd numbers from 15 to 1667
1667 = 15 + (n – 1) × 2
⇒ 1667 = 15 + 2 n – 2
⇒ 1667 = 15 – 2 + 2 n
⇒ 1667 = 13 + 2 n
After transposing 13 to LHS
⇒ 1667 – 13 = 2 n
⇒ 1654 = 2 n
After rearranging the above expression
⇒ 2 n = 1654
After transposing 2 to RHS
⇒ n = 1654/2
⇒ n = 827
Thus, the number of terms of odd numbers from 15 to 1667 = 827
This means 1667 is the 827th term.
Finding the sum of the given odd numbers from 15 to 1667
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 odd numbers from 15 to 1667
= 827/2 (15 + 1667)
= 827/2 × 1682
= 827 × 1682/2
= 1391014/2 = 695507
Thus, the sum of all terms of the given odd numbers from 15 to 1667 = 695507
And, the total number of terms = 827
Since, the average of the given numbers
= Sum of the given numbers/Total number of given numbers
Thus, the average of the given odd numbers from 15 to 1667
= 695507/827 = 841
Thus, the average of the given odd numbers from 15 to 1667 = 841 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 461
(2) Find the average of odd numbers from 13 to 847
(3) What will be the average of the first 4407 odd numbers?
(4) Find the average of odd numbers from 7 to 1175
(5) Find the average of the first 511 odd numbers.
(6) Find the average of the first 3150 odd numbers.
(7) Find the average of the first 1374 odd numbers.
(8) Find the average of even numbers from 4 to 576
(9) What is the average of the first 1470 even numbers?
(10) Find the average of odd numbers from 11 to 1235