Question : Find the average of odd numbers from 13 to 969
Correct Answer 491
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 969
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 969 are
13, 15, 17, . . . . 969
After observing the above list of the odd numbers from 13 to 969 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 969 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 969
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 969
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 969
= 13 + 969/2
= 982/2 = 491
Thus, the average of the odd numbers from 13 to 969 = 491 Answer
Method (2) to find the average of the odd numbers from 13 to 969
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 969 are
13, 15, 17, . . . . 969
The odd numbers from 13 to 969 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 969
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 969
969 = 13 + (n – 1) × 2
⇒ 969 = 13 + 2 n – 2
⇒ 969 = 13 – 2 + 2 n
⇒ 969 = 11 + 2 n
After transposing 11 to LHS
⇒ 969 – 11 = 2 n
⇒ 958 = 2 n
After rearranging the above expression
⇒ 2 n = 958
After transposing 2 to RHS
⇒ n = 958/2
⇒ n = 479
Thus, the number of terms of odd numbers from 13 to 969 = 479
This means 969 is the 479th term.
Finding the sum of the given odd numbers from 13 to 969
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 969
= 479/2 (13 + 969)
= 479/2 × 982
= 479 × 982/2
= 470378/2 = 235189
Thus, the sum of all terms of the given odd numbers from 13 to 969 = 235189
And, the total number of terms = 479
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 969
= 235189/479 = 491
Thus, the average of the given odd numbers from 13 to 969 = 491 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 641
(2) Find the average of even numbers from 12 to 732
(3) Find the average of even numbers from 10 to 352
(4) What is the average of the first 908 even numbers?
(5) Find the average of even numbers from 4 to 1308
(6) Find the average of odd numbers from 11 to 579
(7) Find the average of even numbers from 10 to 1680
(8) What is the average of the first 1780 even numbers?
(9) Find the average of odd numbers from 9 to 1295
(10) Find the average of even numbers from 6 to 646