Question : Find the average of odd numbers from 3 to 1181
Correct Answer 592
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 1181
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 1181 are
3, 5, 7, . . . . 1181
After observing the above list of the odd numbers from 3 to 1181 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1181 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 1181
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1181
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 3 to 1181
= 3 + 1181/2
= 1184/2 = 592
Thus, the average of the odd numbers from 3 to 1181 = 592 Answer
Method (2) to find the average of the odd numbers from 3 to 1181
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 1181 are
3, 5, 7, . . . . 1181
The odd numbers from 3 to 1181 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1181
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 3 to 1181
1181 = 3 + (n – 1) × 2
⇒ 1181 = 3 + 2 n – 2
⇒ 1181 = 3 – 2 + 2 n
⇒ 1181 = 1 + 2 n
After transposing 1 to LHS
⇒ 1181 – 1 = 2 n
⇒ 1180 = 2 n
After rearranging the above expression
⇒ 2 n = 1180
After transposing 2 to RHS
⇒ n = 1180/2
⇒ n = 590
Thus, the number of terms of odd numbers from 3 to 1181 = 590
This means 1181 is the 590th term.
Finding the sum of the given odd numbers from 3 to 1181
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 3 to 1181
= 590/2 (3 + 1181)
= 590/2 × 1184
= 590 × 1184/2
= 698560/2 = 349280
Thus, the sum of all terms of the given odd numbers from 3 to 1181 = 349280
And, the total number of terms = 590
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 3 to 1181
= 349280/590 = 592
Thus, the average of the given odd numbers from 3 to 1181 = 592 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 1252
(2) What is the average of the first 942 even numbers?
(3) Find the average of odd numbers from 11 to 1049
(4) Find the average of the first 673 odd numbers.
(5) Find the average of odd numbers from 5 to 1061
(6) Find the average of odd numbers from 3 to 1053
(7) Find the average of even numbers from 10 to 1010
(8) Find the average of even numbers from 6 to 308
(9) Find the average of even numbers from 4 to 1230
(10) What is the average of the first 1095 even numbers?