Question:
Find the average of odd numbers from 15 to 1161
Correct Answer
588
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 1161
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 1161 are
15, 17, 19, . . . . 1161
After observing the above list of the odd numbers from 15 to 1161 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 1161 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 1161
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1161
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 1161
= 15 + 1161/2
= 1176/2 = 588
Thus, the average of the odd numbers from 15 to 1161 = 588 Answer
Method (2) to find the average of the odd numbers from 15 to 1161
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 1161 are
15, 17, 19, . . . . 1161
The odd numbers from 15 to 1161 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 1161
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 1161
1161 = 15 + (n – 1) × 2
⇒ 1161 = 15 + 2 n – 2
⇒ 1161 = 15 – 2 + 2 n
⇒ 1161 = 13 + 2 n
After transposing 13 to LHS
⇒ 1161 – 13 = 2 n
⇒ 1148 = 2 n
After rearranging the above expression
⇒ 2 n = 1148
After transposing 2 to RHS
⇒ n = 1148/2
⇒ n = 574
Thus, the number of terms of odd numbers from 15 to 1161 = 574
This means 1161 is the 574th term.
Finding the sum of the given odd numbers from 15 to 1161
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 1161
= 574/2 (15 + 1161)
= 574/2 × 1176
= 574 × 1176/2
= 675024/2 = 337512
Thus, the sum of all terms of the given odd numbers from 15 to 1161 = 337512
And, the total number of terms = 574
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 1161
= 337512/574 = 588
Thus, the average of the given odd numbers from 15 to 1161 = 588 Answer
Similar Questions
(1) Find the average of even numbers from 6 to 594
(2) Find the average of odd numbers from 11 to 1399
(3) Find the average of the first 631 odd numbers.
(4) Find the average of the first 3899 even numbers.
(5) Find the average of the first 3536 even numbers.
(6) Find the average of even numbers from 6 to 1560
(7) Find the average of the first 1251 odd numbers.
(8) Find the average of even numbers from 10 to 960
(9) Find the average of the first 2096 odd numbers.
(10) Find the average of even numbers from 12 to 922