Question : Find the average of odd numbers from 13 to 931
Correct Answer 472
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 931
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 931 are
13, 15, 17, . . . . 931
After observing the above list of the odd numbers from 13 to 931 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 931 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 931
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 931
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 931
= 13 + 931/2
= 944/2 = 472
Thus, the average of the odd numbers from 13 to 931 = 472 Answer
Method (2) to find the average of the odd numbers from 13 to 931
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 931 are
13, 15, 17, . . . . 931
The odd numbers from 13 to 931 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 931
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 931
931 = 13 + (n – 1) × 2
⇒ 931 = 13 + 2 n – 2
⇒ 931 = 13 – 2 + 2 n
⇒ 931 = 11 + 2 n
After transposing 11 to LHS
⇒ 931 – 11 = 2 n
⇒ 920 = 2 n
After rearranging the above expression
⇒ 2 n = 920
After transposing 2 to RHS
⇒ n = 920/2
⇒ n = 460
Thus, the number of terms of odd numbers from 13 to 931 = 460
This means 931 is the 460th term.
Finding the sum of the given odd numbers from 13 to 931
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 931
= 460/2 (13 + 931)
= 460/2 × 944
= 460 × 944/2
= 434240/2 = 217120
Thus, the sum of all terms of the given odd numbers from 13 to 931 = 217120
And, the total number of terms = 460
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 931
= 217120/460 = 472
Thus, the average of the given odd numbers from 13 to 931 = 472 Answer
Similar Questions
(1) Find the average of the first 2001 even numbers.
(2) Find the average of even numbers from 4 to 1574
(3) Find the average of even numbers from 8 to 810
(4) Find the average of odd numbers from 9 to 1289
(5) Find the average of odd numbers from 5 to 617
(6) Find the average of the first 3576 even numbers.
(7) Find the average of odd numbers from 15 to 181
(8) Find the average of the first 3015 even numbers.
(9) Find the average of the first 2602 even numbers.
(10) Find the average of even numbers from 8 to 626