Question:
Find the average of odd numbers from 13 to 991
Correct Answer
502
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 991
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 991 are
13, 15, 17, . . . . 991
After observing the above list of the odd numbers from 13 to 991 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 991 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 991
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 991
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 13 to 991
= 13 + 991/2
= 1004/2 = 502
Thus, the average of the odd numbers from 13 to 991 = 502 Answer
Method (2) to find the average of the odd numbers from 13 to 991
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 991 are
13, 15, 17, . . . . 991
The odd numbers from 13 to 991 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 991
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 13 to 991
991 = 13 + (n – 1) × 2
⇒ 991 = 13 + 2 n – 2
⇒ 991 = 13 – 2 + 2 n
⇒ 991 = 11 + 2 n
After transposing 11 to LHS
⇒ 991 – 11 = 2 n
⇒ 980 = 2 n
After rearranging the above expression
⇒ 2 n = 980
After transposing 2 to RHS
⇒ n = 980/2
⇒ n = 490
Thus, the number of terms of odd numbers from 13 to 991 = 490
This means 991 is the 490th term.
Finding the sum of the given odd numbers from 13 to 991
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 13 to 991
= 490/2 (13 + 991)
= 490/2 × 1004
= 490 × 1004/2
= 491960/2 = 245980
Thus, the sum of all terms of the given odd numbers from 13 to 991 = 245980
And, the total number of terms = 490
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 13 to 991
= 245980/490 = 502
Thus, the average of the given odd numbers from 13 to 991 = 502 Answer
Similar Questions
(1) Find the average of even numbers from 10 to 762
(2) Find the average of odd numbers from 13 to 1053
(3) Find the average of odd numbers from 3 to 355
(4) Find the average of even numbers from 6 to 1486
(5) Find the average of even numbers from 12 to 1522
(6) What is the average of the first 682 even numbers?
(7) What is the average of the first 1721 even numbers?
(8) What will be the average of the first 4799 odd numbers?
(9) Find the average of odd numbers from 3 to 1011
(10) Find the average of even numbers from 10 to 1220