Question : Find the average of odd numbers from 3 to 1331
Correct Answer 667
Solution & Explanation
Solution
Method (1) to find the average of the odd numbers from 3 to 1331
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 3 to 1331 are
3, 5, 7, . . . . 1331
After observing the above list of the odd numbers from 3 to 1331 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 3 to 1331 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 3 to 1331
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1331
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 1331
= 3 + 1331/2
= 1334/2 = 667
Thus, the average of the odd numbers from 3 to 1331 = 667 Answer
Method (2) to find the average of the odd numbers from 3 to 1331
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 3 to 1331 are
3, 5, 7, . . . . 1331
The odd numbers from 3 to 1331 form an Arithmetic Series in which
The First Term (a) = 3
The Common Difference (d) = 2
And the last term (ℓ) = 1331
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 1331
1331 = 3 + (n – 1) × 2
⇒ 1331 = 3 + 2 n – 2
⇒ 1331 = 3 – 2 + 2 n
⇒ 1331 = 1 + 2 n
After transposing 1 to LHS
⇒ 1331 – 1 = 2 n
⇒ 1330 = 2 n
After rearranging the above expression
⇒ 2 n = 1330
After transposing 2 to RHS
⇒ n = 1330/2
⇒ n = 665
Thus, the number of terms of odd numbers from 3 to 1331 = 665
This means 1331 is the 665th term.
Finding the sum of the given odd numbers from 3 to 1331
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 1331
= 665/2 (3 + 1331)
= 665/2 × 1334
= 665 × 1334/2
= 887110/2 = 443555
Thus, the sum of all terms of the given odd numbers from 3 to 1331 = 443555
And, the total number of terms = 665
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 1331
= 443555/665 = 667
Thus, the average of the given odd numbers from 3 to 1331 = 667 Answer
Similar Questions
(1) Find the average of even numbers from 8 to 458
(2) What is the average of the first 1199 even numbers?
(3) Find the average of even numbers from 12 to 280
(4) Find the average of odd numbers from 5 to 115
(5) Find the average of the first 532 odd numbers.
(6) Find the average of odd numbers from 15 to 1661
(7) Find the average of the first 2915 odd numbers.
(8) Find the average of even numbers from 12 to 722
(9) Find the average of odd numbers from 13 to 241
(10) Find the average of the first 2250 even numbers.