Question : Find the average of odd numbers from 3 to 1027
Correct Answer 515
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 1027
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 1027 are
3, 5, 7, . . . . 1027
After observing the above list of the odd numbers from 3 to 1027 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1027 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 1027
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1027
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 1027
= 3 + 1027/2
= 1030/2 = 515
Thus, the average of the odd numbers from 3 to 1027 = 515 Answer
Method (2) to find the average of the odd numbers from 3 to 1027
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 1027 are
3, 5, 7, . . . . 1027
The odd numbers from 3 to 1027 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1027
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 1027
1027 = 3 + (n – 1) × 2
⇒ 1027 = 3 + 2 n – 2
⇒ 1027 = 3 – 2 + 2 n
⇒ 1027 = 1 + 2 n
After transposing 1 to LHS
⇒ 1027 – 1 = 2 n
⇒ 1026 = 2 n
After rearranging the above expression
⇒ 2 n = 1026
After transposing 2 to RHS
⇒ n = 1026/2
⇒ n = 513
Thus, the number of terms of odd numbers from 3 to 1027 = 513
This means 1027 is the 513th term.
Finding the sum of the given odd numbers from 3 to 1027
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 1027
= 513/2 (3 + 1027)
= 513/2 × 1030
= 513 × 1030/2
= 528390/2 = 264195
Thus, the sum of all terms of the given odd numbers from 3 to 1027 = 264195
And, the total number of terms = 513
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 1027
= 264195/513 = 515
Thus, the average of the given odd numbers from 3 to 1027 = 515 Answer
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