Question : Find the average of odd numbers from 15 to 961
Correct Answer 488
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 15 to 961
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 15 to 961 are
15, 17, 19, . . . . 961
After observing the above list of the odd numbers from 15 to 961 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 15 to 961 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 15 to 961
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 961
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 961
= 15 + 961/2
= 976/2 = 488
Thus, the average of the odd numbers from 15 to 961 = 488 Answer
Method (2) to find the average of the odd numbers from 15 to 961
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 15 to 961 are
15, 17, 19, . . . . 961
The odd numbers from 15 to 961 form an Arithmetic Series in which
The First Term (a) = 15
The Common Difference (d) = 2
And the last term (ℓ) = 961
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 961
961 = 15 + (n – 1) × 2
⇒ 961 = 15 + 2 n – 2
⇒ 961 = 15 – 2 + 2 n
⇒ 961 = 13 + 2 n
After transposing 13 to LHS
⇒ 961 – 13 = 2 n
⇒ 948 = 2 n
After rearranging the above expression
⇒ 2 n = 948
After transposing 2 to RHS
⇒ n = 948/2
⇒ n = 474
Thus, the number of terms of odd numbers from 15 to 961 = 474
This means 961 is the 474th term.
Finding the sum of the given odd numbers from 15 to 961
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 961
= 474/2 (15 + 961)
= 474/2 × 976
= 474 × 976/2
= 462624/2 = 231312
Thus, the sum of all terms of the given odd numbers from 15 to 961 = 231312
And, the total number of terms = 474
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 961
= 231312/474 = 488
Thus, the average of the given odd numbers from 15 to 961 = 488 Answer
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