Question:
Find the average of odd numbers from 5 to 1325
Correct Answer
665
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 5 to 1325
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 5 to 1325 are
5, 7, 9, . . . . 1325
After observing the above list of the odd numbers from 5 to 1325 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 5 to 1325 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 5 to 1325
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 1325
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 5 to 1325
= 5 + 1325/2
= 1330/2 = 665
Thus, the average of the odd numbers from 5 to 1325 = 665 Answer
Method (2) to find the average of the odd numbers from 5 to 1325
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 5 to 1325 are
5, 7, 9, . . . . 1325
The odd numbers from 5 to 1325 form an Arithmetic Series in which
The First Term (a) = 5
The Common Difference (d) = 2
And the last term (ℓ) = 1325
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 5 to 1325
1325 = 5 + (n – 1) × 2
⇒ 1325 = 5 + 2 n – 2
⇒ 1325 = 5 – 2 + 2 n
⇒ 1325 = 3 + 2 n
After transposing 3 to LHS
⇒ 1325 – 3 = 2 n
⇒ 1322 = 2 n
After rearranging the above expression
⇒ 2 n = 1322
After transposing 2 to RHS
⇒ n = 1322/2
⇒ n = 661
Thus, the number of terms of odd numbers from 5 to 1325 = 661
This means 1325 is the 661th term.
Finding the sum of the given odd numbers from 5 to 1325
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 5 to 1325
= 661/2 (5 + 1325)
= 661/2 × 1330
= 661 × 1330/2
= 879130/2 = 439565
Thus, the sum of all terms of the given odd numbers from 5 to 1325 = 439565
And, the total number of terms = 661
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 5 to 1325
= 439565/661 = 665
Thus, the average of the given odd numbers from 5 to 1325 = 665 Answer
Similar Questions
(1) Find the average of odd numbers from 11 to 511
(2) What is the average of the first 481 even numbers?
(3) Find the average of odd numbers from 5 to 163
(4) Find the average of odd numbers from 15 to 1041
(5) Find the average of odd numbers from 13 to 903
(6) What is the average of the first 1775 even numbers?
(7) Find the average of even numbers from 12 to 800
(8) Find the average of the first 3284 odd numbers.
(9) Find the average of even numbers from 4 to 734
(10) Find the average of the first 3410 even numbers.