Question:
Find the average of odd numbers from 7 to 791
Correct Answer
399
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 7 to 791
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 7 to 791 are
7, 9, 11, . . . . 791
After observing the above list of the odd numbers from 7 to 791 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 7 to 791 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 7 to 791
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 791
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 7 to 791
= 7 + 791/2
= 798/2 = 399
Thus, the average of the odd numbers from 7 to 791 = 399 Answer
Method (2) to find the average of the odd numbers from 7 to 791
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 7 to 791 are
7, 9, 11, . . . . 791
The odd numbers from 7 to 791 form an Arithmetic Series in which
The First Term (a) = 7
The Common Difference (d) = 2
And the last term (ℓ) = 791
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 7 to 791
791 = 7 + (n – 1) × 2
⇒ 791 = 7 + 2 n – 2
⇒ 791 = 7 – 2 + 2 n
⇒ 791 = 5 + 2 n
After transposing 5 to LHS
⇒ 791 – 5 = 2 n
⇒ 786 = 2 n
After rearranging the above expression
⇒ 2 n = 786
After transposing 2 to RHS
⇒ n = 786/2
⇒ n = 393
Thus, the number of terms of odd numbers from 7 to 791 = 393
This means 791 is the 393th term.
Finding the sum of the given odd numbers from 7 to 791
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 7 to 791
= 393/2 (7 + 791)
= 393/2 × 798
= 393 × 798/2
= 313614/2 = 156807
Thus, the sum of all terms of the given odd numbers from 7 to 791 = 156807
And, the total number of terms = 393
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 7 to 791
= 156807/393 = 399
Thus, the average of the given odd numbers from 7 to 791 = 399 Answer
Similar Questions
(1) Find the average of odd numbers from 7 to 655
(2) Find the average of odd numbers from 9 to 1067
(3) Find the average of even numbers from 4 to 336
(4) Find the average of odd numbers from 11 to 1443
(5) Find the average of odd numbers from 3 to 367
(6) Find the average of the first 250 odd numbers.
(7) Find the average of the first 3776 even numbers.
(8) Find the average of the first 778 odd numbers.
(9) Find the average of the first 3812 odd numbers.
(10) Find the average of odd numbers from 9 to 1427