Question : Find the average of odd numbers from 15 to 1149
Correct Answer 582
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1149
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1149 are
15, 17, 19, . . . . 1149
After observing the above list of the odd numbers from 15 to 1149 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1149 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1149
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1149
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 1149
= 15 + 1149/2
= 1164/2 = 582
Thus, the average of the odd numbers from 15 to 1149 = 582 Answer
Method (2) to find the average of the odd numbers from 15 to 1149
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1149 are
15, 17, 19, . . . . 1149
The odd numbers from 15 to 1149 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1149
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 1149
1149 = 15 + (n – 1) × 2
⇒ 1149 = 15 + 2 n – 2
⇒ 1149 = 15 – 2 + 2 n
⇒ 1149 = 13 + 2 n
After transposing 13 to LHS
⇒ 1149 – 13 = 2 n
⇒ 1136 = 2 n
After rearranging the above expression
⇒ 2 n = 1136
After transposing 2 to RHS
⇒ n = 1136/2
⇒ n = 568
Thus, the number of terms of odd numbers from 15 to 1149 = 568
This means 1149 is the 568th term.
Finding the sum of the given odd numbers from 15 to 1149
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 1149
= 568/2 (15 + 1149)
= 568/2 × 1164
= 568 × 1164/2
= 661152/2 = 330576
Thus, the sum of all terms of the given odd numbers from 15 to 1149 = 330576
And, the total number of terms = 568
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 1149
= 330576/568 = 582
Thus, the average of the given odd numbers from 15 to 1149 = 582 Answer
Similar Questions
(1) What is the average of the first 158 odd numbers?
(2) Find the average of even numbers from 10 to 1952
(3) Find the average of odd numbers from 3 to 749
(4) Find the average of the first 2628 even numbers.
(5) Find the average of the first 1281 odd numbers.
(6) Find the average of odd numbers from 9 to 585
(7) Find the average of odd numbers from 13 to 83
(8) Find the average of odd numbers from 15 to 827
(9) Find the average of odd numbers from 11 to 701
(10) Find the average of odd numbers from 3 to 857