Question:
Find the average of odd numbers from 13 to 1375
Correct Answer
694
Solution And Explanation
Solution
Method (1) to find the average of the odd numbers from 13 to 1375
Shortcut Trick to find the average of the given continuous odd numbers
The odd numbers from 13 to 1375 are
13, 15, 17, . . . . 1375
After observing the above list of the odd numbers from 13 to 1375 we find that the difference between two consecutive terms are equal. This means the list of the odd numbers from 13 to 1375 form an Arithmetic Series.
In the Arithmetic Series of the odd numbers from 13 to 1375
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1375
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 13 to 1375
= 13 + 1375/2
= 1388/2 = 694
Thus, the average of the odd numbers from 13 to 1375 = 694 Answer
Method (2) to find the average of the odd numbers from 13 to 1375
Finding the average of given continuous odd numbers after finding their sum
The odd numbers from 13 to 1375 are
13, 15, 17, . . . . 1375
The odd numbers from 13 to 1375 form an Arithmetic Series in which
The First Term (a) = 13
The Common Difference (d) = 2
And the last term (ℓ) = 1375
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 13 to 1375
1375 = 13 + (n – 1) × 2
⇒ 1375 = 13 + 2 n – 2
⇒ 1375 = 13 – 2 + 2 n
⇒ 1375 = 11 + 2 n
After transposing 11 to LHS
⇒ 1375 – 11 = 2 n
⇒ 1364 = 2 n
After rearranging the above expression
⇒ 2 n = 1364
After transposing 2 to RHS
⇒ n = 1364/2
⇒ n = 682
Thus, the number of terms of odd numbers from 13 to 1375 = 682
This means 1375 is the 682th term.
Finding the sum of the given odd numbers from 13 to 1375
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 13 to 1375
= 682/2 (13 + 1375)
= 682/2 × 1388
= 682 × 1388/2
= 946616/2 = 473308
Thus, the sum of all terms of the given odd numbers from 13 to 1375 = 473308
And, the total number of terms = 682
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 13 to 1375
= 473308/682 = 694
Thus, the average of the given odd numbers from 13 to 1375 = 694 Answer
Similar Questions
(1) Find the average of odd numbers from 9 to 109
(2) Find the average of even numbers from 6 to 1618
(3) Find the average of even numbers from 12 to 1324
(4) Find the average of the first 2224 even numbers.
(5) Find the average of the first 2983 odd numbers.
(6) What is the average of the first 1883 even numbers?
(7) What will be the average of the first 4606 odd numbers?
(8) Find the average of odd numbers from 11 to 71
(9) What is the average of the first 1956 even numbers?
(10) Find the average of even numbers from 4 to 316